2.1 Coulomb branch algebras and their partial flag versions

Let $ G $ be a reductive group. Let $ T\subset B$ denote a fixed maximal torus and Borel subgroup, and let W denote the Weyl group. We write $\mathcal K = {\mathbb C}((z)), \mathcal O = {\mathbb C}[[z]] $ , and we consider the groups $ G(\mathcal {K}) = G((z))$ and $G(\mathcal {O}) = G[[z]] $ . We will study the affine Grassmannian $ \mathsf {Gr}_G = G(\mathcal {K})/G(\mathcal {O})$ . Recall that for any dominant coweight $ \lambda $ , the $ G(\mathcal {O}) $ -orbits $ \mathsf {Gr}^\lambda = G(\mathcal {O}) z^\lambda $ partition $\mathsf {Gr}_G $ .

Let $ P$ denote any parabolic subgroup of $ G $ containing T, and L its Levi subgroup. Let $ W_L $ be the Weyl group of $ L$ . There is a corresponding parahoric subgroup $ I_P \subset G(\mathcal {O}) $ , which is defined as the preimage of $ P $ under the evaluation map $ G(\mathcal {O}) \rightarrow G $ . In particular, if $ P = B $ , then $ I_P $ is the usual Iwahori; on the other hand, if $ P = G $ , then $ I_P = G(\mathcal {O}) $ . We have the corresponding partial affine flag variety $ G(\mathcal {K}) / I_P $ .

We fix an extension

$$ \begin{align*}1 \rightarrow G \rightarrow \tilde G \rightarrow F \rightarrow 1, \end{align*} $$

where $ F $ is a torus. Following the physics literature, we call $ F $ the flavour torus of this extension.

Let $ \tilde T $ denote the maximal torus of $ \tilde G $ containing $ T $ . Similarly, for any parabolic subgroup $ P $ of $ G $ , let $\tilde {P}$ be the parabolic subgroup in $\tilde G$ , such that $G\cap \tilde {P}=P$ and $\tilde {I}_P$ its preimage in $\tilde G(\mathcal {O})$ . Note that the Levi $\tilde {L}$ of $\tilde {P}$ is an extension of F by L. We also have that $ L(\mathcal {O}) \subset I_P $ .

Throughout, we will regularly need to use the cohomology rings with complex coefficients of the classifying spaces of these groups, and so we write

and similarly for $ L, F, \tilde G, T $ , etc.

We define the group $\tilde G_{\mathcal {K}}^{\mathcal {O}}$ to be the preimage of $ F(\mathcal {O}) $ under the map $ \tilde G(\mathcal {K}) \rightarrow F(\mathcal {K})$ . This subgroup contains $\tilde G(\mathcal {O})$ , and the quotient $\tilde G_{\mathcal {K}}^{\mathcal {O}}/\tilde G(\mathcal {O})$ is isomorphic to the affine Grassmannian $\mathsf {Gr}_G$ . We have an action of $ {\mathbb C}^\times $ on $ \tilde G_{\mathcal {K}}^{\mathcal {O}} $ by loop rotation, given by $(t\cdot g)(z) = g(tz)$ for $t \in {\mathbb C}^\times $ and $g(z) \in \tilde G_{\mathcal {K}}^{\mathcal {O}}$ . There is an action of the semidirect product $ \tilde G_{\mathcal {K}}^{\mathcal {O}} \rtimes {\mathbb C}^\times $ on $\mathsf {Gr}_G$ .

Let $ N $ be a representation of $ \tilde G $ . We will typically construct $\tilde G$ by starting with a representation of G, and letting $\tilde G$ be the product of G and a torus which acts on N commuting with G; for convenience later, we will not assume this action is faithful.

Following Braverman-Finkelberg-Nakajima [Reference Braverman, Finkelberg and NakajimaBFN18], we will now define the spaces used to construct the Coulomb branch. Let $ N(\mathcal {O}) = N \otimes {\mathbb C}[[z]] $ and $ N(\mathcal {K}) = N \otimes {\mathbb C}((z))$ ; these are naturally representations over $\tilde G(\mathcal {O}) \rtimes {\mathbb C}^\times $ and $ \tilde G_{\mathcal {K}}^{\mathcal {O}} \rtimes {\mathbb C}^\times $ , respectively. We can consider the vector bundle

$$ \begin{align*}T_{G,N} := G(\mathcal{K}) \times_{G(\mathcal{O})} N(\mathcal{O}) = \tilde G_{\mathcal{K}}^{\mathcal{O}} \times_{\tilde G(\mathcal{O})} N(\mathcal{O}) \rightarrow Gr_G. \end{align*} $$

Note that the subscript $G(\mathcal {O})$ (respectively, $\tilde G(\mathcal {O})$ ) indicates the quotient by a natural group action. There is a projection map $ p: T_{G,N} \rightarrow N(\mathcal {K}) $ given by $ [g,v] \mapsto gv $ .

Let

$$ \begin{align*}R_{G,N} = p^{-1}(N(\mathcal{O})) = \{([g], w) \in Gr_G \times N(\mathcal{O}) : w \in g N(\mathcal{O}) \}. \end{align*} $$

Following Braverman-Finkelberg-Nakajima [Reference Braverman, Finkelberg and NakajimaBFN18], we form the convolution algebra

$$ \begin{align*}\mathcal A_{\hbar}(G,N) := H_*^{\tilde G(\mathcal{O}) \rtimes {\mathbb C}^\times}(R_{G,N}).\end{align*} $$

We will use $\hbar $ throughout for the equivariant parameter corresponding to the loop ${\mathbb C}^\times $ -action. We include it in the notation here to emphasize that we have left it as a free variable, whereas later, we will usually consider the quotient where we set $\hbar =1$ . A great deal of care is required to define the equivariant Borel-Moore homology for an infinite-dimensional space and group (see [Reference Braverman, Finkelberg and NakajimaBFN18, Section 2(ii)] for a detailed discussion). We will refer to $\mathcal A_{\hbar }(G,N)$ as the spherical Coulomb branch algebra.

We will also need the partial flag variety version of the BFN algebra.

Definition 2.2. Let $ P $ be a parabolic subgroup of $ G$ . We define

$$ \begin{align*} T_{G,N}^P& := G(\mathcal{K}) \times_{I_P} N(\mathcal{O}) = \tilde G_{\mathcal{K}}^{\mathcal{O}} \times_{\tilde{I}_P} N(\mathcal{O})\xrightarrow{p_P} N(\mathcal{K})\\ R_{G,N}^P &:= p_P^{-1}(N(\mathcal{O})) = \{([g], w) \in (G(\mathcal{K})/ I_P) \times N(\mathcal{O}) : w \in g N(\mathcal{O}) \}. \end{align*} $$

The parabolic Coulomb branch algebra is the convolution algebra

$$ \begin{align*} \mathcal A_{\hbar}^P(G,N) := H_*^{\tilde{I}_P \rtimes {\mathbb C}^\times}(R_{G,N}^P). \end{align*} $$

These algebras depend on the choice of $\tilde G$ , but we leave this implicit in the notation. In the special case where $ P = B$ , then we will call $ \mathcal A^B_{\hbar } $ the Iwahori Coulomb branch algebra. The spherical Coulomb branch algebra is an example of a principal Galois order, as defined, that is in [Reference Futorny, Grantcharov, Ramirez and ZadunaiskyFGRZ20]. The idea of replacing principal Galois orders by flag orders (defined in [Reference WebsterWeba, Lemma 2.5]) played an important role in [Reference WebsterWeba, Reference WebsterWebb]; the algebra $\mathcal A^B_{\hbar }$ is an example of a flag order in this case.

Note that the space $ R_{G,N}^P$ contains a copy of $(G(\mathcal {O})/I_P)\times N(\mathcal {O})\cong G/P\times N(\mathcal {O})$ . We have vector space isomorphisms

(2.1) $$ \begin{align} H_*^{{\tilde{I}_P \rtimes {\mathbb C}^\times}}(G/P\times N(\mathcal{O}))\cong H_*^{\tilde{P} \times {\mathbb C}^\times}(G/P)\cong H_*^{\tilde G \times {\mathbb C}^\times}((G/P)^2) \end{align} $$

(here, we use $ G/P = \tilde G / \tilde P $ to get the action of $ \tilde G $ on $G/P $ ).

Now, $ H_*^{\tilde G \times {\mathbb C}^\times }((G/P)^2) $ has a convolution structure of its own, and Poincaré duality shows that it is a matrix algebra:

When $ P = B $ , then this is the nilHecke algebra of $ W$ .

Lemma 2.3. The inclusion $ (G(\mathcal {O})/I_P)\times N(\mathcal {O}) \hookrightarrow R_{G,N}^P $ induces an algebra map

$$ \begin{align*}H_*^{\tilde G \times {\mathbb C}^\times }((G/P)^2)\to \mathcal A_{\hbar}^P(G,N).\end{align*} $$

Let $e^{\prime } \in H_*^{\tilde G \times {\mathbb C}^\times }((G/P)^2)=H_*^{\tilde {P} \times {\mathbb C}^\times }(G/P)$ be the primitive idempotent in this matrix algebra that projects to the W-invariants. We can formulate the usual Abelianization isomorphism (see, for example, Proposition 1 in [Reference BrionBri98]) as the statement that for any $\tilde G$ -space X, we have $e^{\prime }H_*^{\tilde {P}}(X)\cong H_*^{\tilde G}(X)$ , where $H_*^{\tilde {P} \times {\mathbb C}^\times }(G/P)$ acts by convolution.

Applying this fact, we find that:

Proposition 2.4. For any parabolic $ P \subset G$ , we have isomorphisms

(2.2) $$ \begin{align} \mathcal A_{\hbar}^P e^{\prime} \cong H_*^{\tilde{I}_P \rtimes {\mathbb C}^\times}(R_{G,N}) \qquad e^{\prime}\mathcal A_{\hbar}^P \cong H_*^{\tilde G(\mathcal{O}) \rtimes {\mathbb C}^\times}(R_{G,N}^P)\qquad \mathcal A_{\hbar} \cong e^{\prime}\mathcal A_{\hbar}^P e^{\prime} .\end{align} $$

The bimodules $ \mathcal A_{\hbar }^P e^{\prime } $ and $ e^{\prime }\mathcal A_{\hbar }^P $ define a Morita equivalence between $ \mathcal A_{\hbar }^P $ and $\mathcal A_{\hbar }$ .

The first two isomorphisms of (2.2) are related by the map from $\tilde {I}_P$ -orbits on $R_{G,N}$ to $\tilde G(\mathcal {O})$ orbits on $R_{G,N}^P$ sending $([g],v)\mapsto ([g^{-1}],g^{-1}v)$ .

The algebra $\mathcal A_{\hbar }(G,N)$ contains a subalgebra given by the equivariant parameters

Braverman et al. [Reference Braverman, Finkelberg and NakajimaBFN18, Section 3(vi)] call this the Cartan subalgebra, but we prefer to call this the Gelfand-Tsetlin subalgebra, since it is a generalization of this subalgebra in $U(\mathfrak {gl}_n)$ . Similarly, $ \mathcal A_{\hbar }^P(G,N) $ contains .

The Gelfand-Tsetlin algebra of $ \mathcal A_{\hbar }(G,N) $ contains the subalgebra , which is central in $\mathcal A_{\hbar }(G,N)$ (here, $ \mathfrak f $ is the Lie algebra of our flavour torus $ F $ ). In fact, an application of [Reference Futorny and OvsienkoFO10, Theorem 4.1(4)], using the fact that $\mathcal A_{\hbar }(G,N)$ is a Galois order as shown in [Reference WebsterWeba, Theorem B], shows that $ Z $ is the full centre of $\mathcal A_{\hbar }(G,N)$ .

2.2 Inclusion of Coulomb branch algebras

One advantage of considering these parabolic Coulomb branch algebras is that they allow us to study the relation with Coulomb branch algebras defined by Levi subgroups.

The inclusion $ L(\mathcal {K}) \subset G(\mathcal {K})$ gives rise to an inclusion $ \mathsf {Gr}_L \hookrightarrow G(\mathcal {K}) / I_P $ . Moreover, it is easy to see that the restriction of $R^P_{G,N} $ to $ \mathsf {Gr}_L $ coincides with $ R_{L,N}$ . This leads to the following result.

Proof. The inclusion $ R_{L,N} \subset R^P_{G,N} $ gives an inclusion

(2.3) $$ \begin{align} \iota \colon H_*^{L(\mathcal{O})\rtimes {\mathbb C}^\times}(R_{L,N}) \hookrightarrow H_*^{L(\mathcal{O})\rtimes {\mathbb C}^\times}(R_{G,N}^P). \end{align} $$

To see the compatibility of this inclusion with multiplication, we use an argument similar to that in [Reference Braverman, Finkelberg and NakajimaBFN18, Section 5(ii)]. Consider the analogue of the diagram [Reference Braverman, Finkelberg and NakajimaBFN18, (3.2)]:

where $ p : R^P_{G,N} \times L(\mathcal {K}) \rightarrow R^P_{G,N} \times T_{L,N} $ is given by $ ([g_2], w, g_1) \mapsto ([g_2], w, [g_1, w])$ . This diagram defines a right action of $\mathcal A_{\hbar }(L,N)$ on $\mathcal A_{\hbar }^P(G,N)$ . In fact, this diagram is a special case of the auxiliary action diagram (27) from [Reference Hilburn, Kamnitzer and WeekesHKW], where $ Z = R^P_{G,N} $ and $ G = L$ , except with all factors reversed. Thus, from [Reference Hilburn, Kamnitzer and WeekesHKW, Proposition 4.13], we can deduce that it defines a right module structure of $ \mathcal A_{\hbar }(L,N)$ on $\mathcal A_{\hbar }^P(G,N)$ commuting with left multiplication.

As in [Reference Braverman, Finkelberg and NakajimaBFN18, Lemma 5.7(1)], we can see that $1\star b =\iota (b)$ . More generally, we must have $a\star b=a\iota (b)$ . This shows that for any $b,b^{\prime }\in \mathcal A_{\hbar }(L,N)$ , we have:

$$\begin{align*}\iota(bb^{\prime})=1\star(bb^{\prime})=(1\star b)\star b^{\prime}=\iota(b)\star b^{\prime}=\iota(b)\iota(b^{\prime}).\end{align*}$$

Thus, (2.3) is an algebra map.

The inclusion $ L(\mathcal {O}) \subset I_P $ leads to an isomorphism

$$ \begin{align*} H_*^{L(\mathcal{O})\rtimes {\mathbb C}^\times}(R_{G,N}^P) \cong H_*^{I_P \rtimes {\mathbb C}^\times}(R_{G,N}^P). \end{align*} $$

Composing this with (2.3) gives the desired injection.

2.3 Abelian theories and monopole operators

Let $ \nu : {\mathbb C}^\times \rightarrow T $ be a central coweight. Since $ \nu $ is central, $ z^\nu \in \mathsf {Gr}_G$ is a $ G(\mathcal {O})$ orbit. We let $r_{\nu }\in \mathcal A_{\hbar }(G,N)$ be the homology class of the preimage of $z^\nu $ in $R_{G,N}$ . These elements $r_\nu $ are called monopole operators.

When G is Abelian, all coweights are central, and the elements $r_\nu $ form a basis for $\mathcal A_{\hbar }(G, N)$ as a left (or right) module over , where $\nu $ runs over all coweights of G. Their relations are known explicitly by [Reference Braverman, Finkelberg and NakajimaBFN18, Section 4(iii)]:

(2.4) $$ \begin{align} r_\xi r_\nu = \prod_{\langle\mu,\xi\rangle> 0 > \langle \mu,\nu\rangle} \prod_{j=1}^{d(\langle \mu, \xi\rangle, \langle \mu,\nu\rangle)} \big(\mu + (\langle\mu, \xi\rangle - j) \hbar \big ) \prod_{\langle\mu,\xi\rangle < 0 < \langle \mu,\nu\rangle} \prod_{j=0}^{d(\langle \mu, \xi\rangle, \langle \mu,\nu\rangle)-1} \big(\mu + (\langle\mu, \xi\rangle + j) \hbar \big ) r_{\xi+\nu}. \end{align} $$

Here, the first and third products range over weights $\mu $ of N, with multiplicity. These are weights for the action of $ \tilde G$ , and the products above lie in the Gelfand-Tsetlin subalgebra . Also $d: {\mathbb Z} \times {\mathbb Z} \rightarrow {\mathbb Z}_{\geq 0}$ is the function defined by

$$ \begin{align*}d(a,b) = \left\{ \begin{array}{cl} 0, & \text{if } a,b \text{ have the same sign,} \\ \min\{|a|, |b|\}, & \text{if } a,b \text{ have different signs.} \end{array} \right. \end{align*} $$

It will also be useful to have an inverted version of these formulas:

(2.5) $$ \begin{align} r_\xi ^{-1}r_{\nu}=\prod_{\langle\mu,\xi\rangle> 0 > \langle \mu,\nu-\xi\rangle} \prod_{j=1}^{d(\langle \mu, \xi\rangle, \langle \mu,\nu-\xi\rangle)} \frac{1}{\mu - j \hbar} \prod_{\langle\mu,\xi\rangle < 0 < \langle \mu,\nu-\xi\rangle} \prod_{j=0}^{d(\langle \mu, \xi\rangle, \langle \mu,\nu-\xi \rangle)-1} \frac{1}{\mu + j \hbar } r_{\nu-\xi}. \end{align} $$

This version follows by rearranging (2.4) for the product $r_\xi r_{\nu -\xi }$ , using in addition the fact that for any weight , we have

$$ \begin{align*}r_\xi \mu = (\mu + \langle \mu,\xi \rangle \hbar) r_\xi \end{align*} $$

by [Reference Braverman, Finkelberg and NakajimaBFN18, (4.8)]. Note that this implies $r_\xi ^{-1} \mu = (\mu - \langle \mu ,\xi \rangle \hbar ) r_\xi ^{-1}$ .

We now recall some algebra homomorphisms defined in [Reference Braverman, Finkelberg and NakajimaBFN18], which will play an important role in this paper.

For general G, there is an Abelianization map $(\iota _\ast )^{-1} : \mathcal A_{\hbar }(G,N) \hookrightarrow \mathcal A_{\hbar }(T,N)_{loc}$ described in [Reference Braverman, Finkelberg and NakajimaBFN18, Remark 5.23]. Here, $\mathcal A_{\hbar }(T,N)_{loc}$ denotes the localization of $\mathcal A_{\hbar }(T,N)$ at the multiplicative set generated by $\hbar , \alpha + m\hbar $ , where $\alpha $ runs over roots of G and $m\in {\mathbb Z}$ .

Next, recall from [Reference Braverman, Finkelberg and NakajimaBFN18, Remark 5.14] that for any two representations $N_1, N_2$ , there is a natural injective map $ \mathcal A_{\hbar }(G,N_1 \oplus N_2) \hookrightarrow \mathcal A_{\hbar }(G, N_2)$ . For $ G $ Abelian, this map is given by [Reference Braverman, Finkelberg and NakajimaBFN18, Section 4(vi)]:

(2.6) $$ \begin{align} r_\nu \mapsto \prod_{\langle \mu, \nu\rangle < 0} \prod_{j=\langle \mu,\nu\rangle}^{-1} (\mu + j \hbar) \ r_\nu. \end{align} $$

Here, the product is over the weights of $ N_1 $ . Also, since it is potentially confusing, we note that we have written $r_\nu $ for the monopole operators in $\mathcal A_{\hbar }(G, N_1\oplus N_2)$ and $\mathcal A_{\hbar }(G,N_2)$ , respectively.

Finally, assume there is a coweight $\wp \colon {\mathbb C}^*\hookrightarrow \tilde {T}$ which acts on $N_2$ by scalar multiplication of weight 1 and on $N_1$ with weight $0$ ; this is always possible if we extend the flavour torus F. By [Reference Braverman, Finkelberg and NakajimaBFN18, Section 6(viii)], there is a ‘Fourier transform’ isomorphism

(2.7) $$ \begin{align} \mathcal A_{\hbar}(G, N_1 \oplus N_2) \stackrel{\sim}{\longrightarrow} \mathcal A_{\hbar}(G, N_1 \oplus N_2^\ast), \end{align} $$

which is the identity on the Gelfand-Tsetlin subalgebra . In the Abelian case, this isomorphism is defined by the map

(2.8) $$ \begin{align} r_{\nu}\mapsto (-1)^{\delta(\nu)}r_{\nu}\qquad \mu \mapsto \mu+\hbar \langle \mu, \wp\rangle, \end{align} $$

where $\delta (\nu )=\sum _{\langle \mu ,\nu \rangle>0}\langle \mu ,\nu \rangle $ . In this sum, $\mu $ ranges over weights of $N_2$ (if we use the conventions of [Reference Braverman, Finkelberg and NakajimaBFN18], as discussed in Remark 2.6, then this shift by $\wp $ is unnecessary).

For general G, the isomorphism is defined using the Abelian case, via the Abelianization map [Reference Braverman, Finkelberg and NakajimaBFN18, Lemmas 5.9–5.10].

2.4 Passing to invariants

For $ \lambda $ a dominant coweight of $ G$ , let $ R_{G,N}^\lambda $ denote the preimage of the $G(\mathcal {O})$ -orbit closure $ \overline {\mathsf {Gr}^\lambda } $ under the map $ R_{G,N} \rightarrow \mathsf {Gr}_G $ . We write $ \mathcal A_{\hbar }(G, N)^\lambda $ for the subspace of the Coulomb branch algebra coming from the homology of $ R_{G,N}^\lambda $ . We can restrict the injective algebra map $ \mathcal A_{\hbar }(G,N_1 \oplus N_2) \hookrightarrow \mathcal A_{\hbar }(G, N_2)$ to an injective linear map $ \mathcal A_{\hbar }(G,N_1 \oplus N_2)^\lambda \hookrightarrow \mathcal A_{\hbar }(G, N_1)^\lambda $ .

Proof. The map $\mathcal A_{\hbar }(G,N_1 \oplus N_2)^\lambda \hookrightarrow \mathcal A_{\hbar }(G, N_1)^\lambda $ comes from pullback along the map

$$ \begin{align*}R_{G,N_1 \oplus N_2}^\lambda \rightarrow R_{G,N_1}^\lambda.\end{align*} $$

However, by the hypothesis, this map is a vector bundle (with fibre over $([g],w) \in R_{G,N_1}^\lambda $ being $g(N_2 \otimes \mathcal {O}) $ ).

Definition 2.8. For any coweight $\xi \colon {\mathbb C}^\times \to T$ , we let $N_0^{\xi }, N_\pm ^{\xi }, N_{\geq }^{\xi }, N_{\leq }^{\xi }$ be the sum of weight spaces where the weight of $\xi $ is zero, positive, negative, nonnegative or nonpositive, respectively.

Note that we always have

$$\begin{align*}N = N^{\xi}_- \oplus N^{\xi}_0 \oplus N^{\xi}_+= N^{\xi}_{\leq} \oplus N^{\xi}_+= N^{\xi}_- \oplus N^{\xi}_{\geq}.\end{align*}$$

We will assume that the coweight $\xi $ is central. Recall the monopole operator $r_\xi $ from Section 2.3.

Theorem 2.9. Assume that $ N^{\xi }_- = 0 $ . The natural map $ \mathcal A_{\hbar }(G, N) \hookrightarrow \mathcal A_{\hbar }(G, N^{\xi }_0) $ gives an isomorphism

$$ \begin{align*}\mathcal A_{\hbar}(G, N)[r_\xi^{-1}] \cong \mathcal A_{\hbar}(G, N^{\xi}_0).\end{align*} $$

For the proof, we appeal to the following basic fact about Ore localizations:

2.5 Hamiltonian reduction

Note that the operations we’ve discussed thus far only relate groups with the same rank, so the dimension of the Coulomb branch will be unchanged. When we change the matter from N to $N^{\xi }_0$ , the action of the central subgroup $ {\mathbb C}^\times _\xi $ becomes trivial. So we can consider $N^\xi _0 $ as a representation of the quotient group $ G/{\mathbb C}^\times _\xi $ . This invites us to consider the relationship between $\mathcal A_{\hbar }(G, N^{\xi }_0)$ and $\mathcal A_{\hbar }(G/{\mathbb C}^\times _\xi , N^{\xi }_0)$ , which decreases the dimension of the Coulomb branch by 2.

Let A be an algebra, and $b\in A$ . Recall that the quantum Hamiltonian reduction of $ A$ by b (at level 1) is defined in two stages. First, we form the right A–module $ A / (b -1) A$ . Then, we construct the quantum Hamiltonian reduction by considering

$$ \begin{align*}A \mathbin{/\mkern-6mu/}_1 b := \operatorname{End}_A\big(A / (b-1)A\big) \cong \big\{ [a] \in A / (b-1)A \ \mid \ a(b-1) \in (b-1)A \big\}. \end{align*} $$

Thought of as an endomorphism ring, $A \mathbin {/\mkern -6mu/}_1 b$ has a natural algebra structure, which may equivalently be defined on equivalence classes by $[a_1]\cdot [a_2] = [a_1 a_2]$ . Geometrically, this algebra is the quantization of the operation of passing to the level set $b=1$ , and then dividing by the flow of the Hamiltonian vector field associated to $ b $ .

Theorem 2.11. Assume that $ \xi : {\mathbb C}^\times \rightarrow G $ is central and primitive (i.e. it is not an integer multiple of any other cocharacter), and that $ {\mathbb C}^\times _\xi $ acts trivially on $ N $ . Then we have

$$ \begin{align*} \mathcal A_{\hbar}(G, N) \mathbin{/\mkern-6mu/}_1 r_\xi \cong \mathcal A_{\hbar}(G / {\mathbb C}^\times_\xi, N). \end{align*} $$

Proof. For the purposes of this proof, let us write $ G^{\prime } = G/{\mathbb C}^\times _\xi $ . Consider $ R_{G^{\prime }, N}$ . This space has an action of $ G(\mathcal {O}) $ which factors through the map $ G(\mathcal {O}) \rightarrow G^{\prime }(\mathcal {O})$ . On the one hand,

$$ \begin{align*} H_*^{G(\mathcal{O}) \rtimes {\mathbb C}^\times}(R_{G^{\prime}}, N) \cong \mathcal A_{\hbar}(G, N) / (r_\xi - 1) \mathcal A_{\hbar}(G,N)\end{align*} $$

as a module over $ \mathcal A_{\hbar }(G, N) $ . To see this, we note that $ R_{G^{\prime }, N} = R_{G,N} / {\mathbb Z} $ , where the generator of $ {\mathbb Z} $ acts by translation by the element $ z^\xi $ , since $\xi $ is primitive.

On the other hand,

as a $ \mathcal A_{\hbar }(G^{\prime }, N) $ module. Therefore, we have that

Note that , where a is a character of the Lie algebra $\mathfrak {g}$ splitting the derivative of $\xi $ . The action of $r_\xi $ on the RHS above commutes with , and satisfies $[r_\xi ,a]=r_{\xi }$ . This shows that the kernel of $r_{\xi }-1$ is $\mathcal A_{\hbar }(G^{\prime }, N)$ .

Remark 2.13. One can also consider the ‘right’ quantum Hamiltonian reduction

$$ \begin{align*} \operatorname{End}_A\big( A/ A(b-1)\big)^{op} \ \cong \ \big\{ [a] \in A / A (b-1) \ \mid \ (b-1)a \in A(b-1) \big\}. \end{align*} $$

With the same assumptions as in the previous theorem, the right Hamiltonian reduction of $\mathcal A_{\hbar }(G,N)$ by $r_\xi $ is also isomorphic to $\mathcal A_{\hbar }(G / {\mathbb C}^\times _\xi , N)$ .

2.6 Combining all the steps

Now, we will see how to combine the above results.

Let $ G $ be a reductive group, $ N $ a representation, $ \xi : {\mathbb C}^\times \rightarrow T $ any coweight. Let $ P, L $ be the parabolic and Levi subgroups corresponding to $ \xi $ , and let $ N^{\xi }_0 $ be the invariants for the action of $ {\mathbb C}^\times _\xi $ on $ N $ .

Theorem 2.14. The algebras

$$ \begin{align*}\mathcal A_{\hbar}(G,N),\ \mathcal A_{\hbar}^P(G,N),\ \mathcal A_{\hbar}(L,N),\ \mathcal A_{\hbar}(L, N^{\xi}_0), \text{ and }\ \mathcal A_{\hbar}(L/{\mathbb C}^\times_\xi, N^{\xi}_0)\end{align*} $$

are related as follows.

1. 1. There is a Morita equivalence between

   $$ \begin{align*} \mathcal A_{\hbar}(G,N) \text{ and } \mathcal A_{\hbar}^P(G,N). \end{align*} $$

2. 2. There is an inclusion of algebras

   $$ \begin{align*}\mathcal A_{\hbar}(L,N) \hookrightarrow \mathcal A_{\hbar}^P(G,N). \end{align*} $$

3. 3. There is an isomorphism

   $$ \begin{align*} \mathcal A_{\hbar}(L,N)[r_\xi^{-1}] \cong \mathcal A_{\hbar}(L, N^{\xi}_0). \end{align*} $$

4. 4. If $\xi $ is primitive, there is an isomorphism

   $$ \begin{align*} \mathcal A_{\hbar}(L,N^{\xi}_0) \mathbin{/\mkern-6mu/}_1 r_\xi \cong \mathcal A_{\hbar}(L/{\mathbb C}^\times_\xi, N^{\xi}_0). \end{align*} $$

All these maps are compatible with the maps between the Gelfand-Tsetlin subalgebras

3.1 Quiver data

Let $\Gamma $ be a quiver with vertex set I and edge set $\mathsf {E}(\Gamma )$ , and dimension vectors $\mathbf {v},\mathbf {w}\colon I\to {\mathbb Z}_{\geq 0}$ (we allow loops and multiple edges). For an edge $ e \in \mathsf {E}(\Gamma )$ , we write $ h(e) $ for its head and $ t(e) $ for its tail.

Consider the group and representation

$$\begin{align*}N=\bigoplus_{e \in \mathsf{E}(\Gamma)}\operatorname{Hom}({\mathbb C}^{v_{t(e)}},{\mathbb C}^{v_{h(e)}})\oplus \bigoplus_{i\in I}\operatorname{Hom}({\mathbb C}^{v_i},{\mathbb C}^{w_i})\qquad G=\prod_{i\in I} GL(v_i).\end{align*}$$

We consider also the larger group

$$ \begin{align*}\tilde G := G \times \prod_{i\in I} ({\mathbb C}^\times)^{w_i} \times ({\mathbb C}^\times)^{\mathsf{E}(\Gamma)},\end{align*} $$

where the middle factor is the product of the diagonal matrices inside each $ GL(w_i) $ . We have a (nonfaithful) action of $\tilde G$ on N. We’ll use $\mathcal A_{\hbar }(\mathbf {v},\mathbf {w}) := \mathcal A_{\hbar }(G, N)$ to denote the Coulomb branch algebra attached to these dimension vectors.

This is slightly cleaner if we use the ‘Crawley-Boevey [Reference Crawley-BoeveyCB01] trick’ of adding a new vertex $\infty $ and $w_i$ new edges from i to $\infty $ to make a larger quiver $\Gamma ^{\mathbf {w}}$ . We extend $ \mathbf {v} $ to this new vertex by setting $ v_\infty = 1 $ . Then, we can think of an element of N as a representation of $\Gamma ^{\mathbf {w}}$ , using $ \operatorname {Hom}({\mathbb C}^{v_i}, {\mathbb C}^{w_i}) = \operatorname {Hom}({\mathbb C}^{v_i}, {\mathbb C}^{v_\infty })^{\oplus w_i} $ , so $ N = \bigoplus _{e \in \mathsf {E}(\Gamma ^{\mathbf {w}})} \operatorname {Hom}({\mathbb C}^{v_{t(e)}},{\mathbb C}^{v_{h(e)}}) $ . Also, from this perspective, $\tilde G$ is obtained from G by adding a scaling along each edge of $\Gamma ^{\mathbf {w}}$ .

3.2 Relating algebras

As in the general case, we want to consider a coweight $\xi \colon {\mathbb C}^\times \to T$ . This is the same as choosing a $ {\mathbb Z}$ -grading on the vector spaces ${\mathbb C}^{v_i}$ ; to correct some later sign issues, we let the degree k elements be those with weight $-k$ . Thus, for each $p\in {\mathbb Z}$ , we have a dimension vector $v_i^{(p)}$ , such that $v_i=\sum _{p\in {\mathbb Z}}v_i^{(p)}$ .

In this case, L is the set of grading preserving automorphisms of these vector spaces, so $L=\prod _{p\in {\mathbb Z}} L^{(p)}$ , where $L^{(p)}=\prod _{i\in I}GL(v_i^{(p)})$ . The subspace $N^{\xi }_0$ is just the grading preserving quiver representations, where ${\mathbb C}^{w_i}$ is given degree 0. That is, $N^{\xi }_0=\bigoplus _{p \in {\mathbb Z}} N^{(p)}$ , where

$$\begin{align*}N^{(p)}= \begin{cases} \displaystyle \bigoplus_{e \in \mathsf{E}(\Gamma)}\operatorname{Hom}({\mathbb C}^{v_{t(e)}^{(p)}},{\mathbb C}^{v_{h(e)}^{(p)}}) & p\neq 0\\ \displaystyle\bigoplus_{e \in \mathsf{E}(\Gamma)}\operatorname{Hom}({\mathbb C}^{v_{t(e)}^{(p)}},{\mathbb C}^{v_{h(e)}^{(p)}})\oplus \bigoplus_{i}\operatorname{Hom}({\mathbb C}^{v_i^{(p)}},{\mathbb C}^{w_i}) & p=0. \end{cases}\end{align*}$$

By [Reference Braverman, Finkelberg and NakajimaBFN18, (3(vii).(a))], we immediately deduce the following.

Proposition 3.1. We have an isomorphism of algebras

$$\begin{align*}\mathcal A_{\hbar}(L, N^{\xi}_0)\cong \cdots \otimes \mathcal A_{\hbar}(\mathbf{v}^{(-1)}, 0)\otimes \mathcal A_{\hbar}(\mathbf{v}^{(0)}, \mathbf{w})\otimes \mathcal A_{\hbar}(\mathbf{v}^{(1)}, 0)\otimes\cdots,\end{align*}$$

where each factor on the RHS is again the Coulomb branch of a quiver gauge theory of the same underlying quiver, all but one of them unframed.

Thus, for a quiver gauge theory, Theorem 2.14 relates the algebra $\mathcal A_{\hbar }(\mathbf {v},\mathbf {w})$ to the tensor product appearing in this proposition.

3.3 Truncated shifted Yangians

Assume that $\Gamma $ is a Dynkin quiver, and let $ \mathfrak {g}_\Gamma $ be the corresponding simply-laced simple Lie algebra. The dimension vectors $\mathbf w, \mathbf v$ , encode a pair of coweights $\lambda , \mu $ , for $ \mathfrak {g}_\Gamma $ , where $ \lambda = \sum w_i \varpi _i $ and $ \lambda - \mu = \sum v_i \alpha _i $ .

Let $ Y^\lambda _\mu $ denote the corresponding truncated shifted Yangian (with formal parameters and with $ \hbar $ ) as defined in appendix B(viii) of [Reference Braverman, Finkelberg and NakajimaBFN19]. In [Reference WeekesWee], the fourth author proved that there is an isomorphism $ \mathcal A_{\hbar }(\mathbf {v}, \mathbf {w}) \cong Y^\lambda _\mu $ , building on the results in appendix B of [Reference Braverman, Finkelberg and NakajimaBFN19].

The splitting $v_i=\sum _{p\in {\mathbb Z}} v_i^{(p)} $ gives us a list of $ \mu ^{(p)} $ , where $ \mu ^{(p)} = - \sum v_i^{(p)} \alpha _i$ for $ p \ne 0 $ and $ \lambda - \mu ^{(0)} = \sum v_i^{(0)} \alpha _i $ (in particular, we have $ \mu = \sum \mu ^{(p)} $ ). Thus, Theorem 2.14 and Proposition 3.1 relate $Y^\lambda _\mu $ with $ Y^\lambda _{\mu ^{(0)}} \otimes \bigotimes _{p \ne 0 } Y^0_{\mu ^{(p)}} $ . For example, we can relate $ Y^\lambda _{\mu ^{(0)} + \mu ^{(1)}} $ with $ Y^\lambda _{\mu ^{(0)}} \otimes Y^0_{\mu ^{(1)}}$ .

An important special case is when we take $ \mu ^{(1)} = -\alpha _i $ for some $ i$ and all other $ \mu ^{(p)} = 0 $ . This corresponds to taking $ \xi $ to be the ith standard coweight $ \xi = \varpi _{i,1} $ (i.e. we map $ {\mathbb C}^\times $ into $ G $ by sending it to the upper left matrix slot of $ GL(v_i)$ ). In this case, $( L/{\mathbb C}^\times _\xi , N^{\xi }_0) $ is the quiver gauge theory corresponding to $ \lambda $ and $\mu + \alpha _i $ , so $ \mathcal A_{\hbar }(L/{\mathbb C}^\times _\xi , N^{\xi }_0) \cong Y^\lambda _{\mu + \alpha _i}$ . Thus, Theorem 2.14 relates $Y^\lambda _\mu $ and $Y^\lambda _{\mu + \alpha _i}$ . In Section 9.2, this will allow us to construct functors between category $\mathcal O $ for these algebras, giving a categorical $ \mathfrak {g}_\Gamma $ -action.

4.1 Gelfand-Tsetlin modules

We wish to understand the representation theory of the algebra $\mathcal A_{\hbar }(G,N)$ , following the approach of [Reference WebsterWebb, Reference Kamnitzer, Tingley, Webster, Weekes and YacobiKTW⁺19b, Reference WebsterWeba]. In particular, we will focus on the Gelfand-Tsetlin modules.

Now, and for the remainder of the paper, we specialize to $ \hbar = 1 $ , and we will write $ \mathcal A(G,N) $ for the result of this specializationFootnote ¹ . Later in the paper, we will further specialize at a point $ \varphi \in \mathfrak f = \operatorname {MaxSpec} Z $ , and consider $ \mathcal A_\varphi (G,N) = \mathcal A(G,N)/ \mathcal A(G,N) \mathsf m_\varphi $ , where $\mathsf {m}_\varphi \subset Z$ is the corresponding maximal ideal.

Given , let be the corresponding maximal ideal. Consider the weight functors defined by

The reader might reasonably be concerned about the fact that this is a generalized eigenspace. In this paper, we will always want to consider these, and thus will omit ‘generalized’ before instances of ‘weight.’ These spaces are called Gelfand-Tsetlin (GT) weight spaces.

Definition 4.1. An $\mathcal A(G,N)$ -module M is called a (Z -semisimple) Gelfand-Tsetlin module if it is finitely generated, , and the centre $ Z $ acts semisimply on $ M $ . For the rest of the paper, we let denote the restriction of the weight functor to the category of Z-semisimple Gelfand-Tsetlin modules.

The support of a Gelfand-Tsetlin module is the set

4.2 Morphisms between weight functors

Gelfand-Tsetlin modules can be classified using the approach of [Reference Drozd, Futorny and OvsienkoDFO94], which is based on understanding the space of natural transformations . Observe that

for $ \gamma , \gamma ^{\prime } \in \tilde {\mathfrak t}/W $ with common image $ \varphi \in \mathfrak f $ . If $ \gamma , \gamma ^{\prime } $ don’t have the same image in $\mathfrak {f}$ , then .

The space has a natural weak operator topology, where a sequence converges if and only if it is eventually constant on for all M. This is the same as the inverse limit topology.

These spaces can be organized into the algebra , or alternatively, into the category whose objects are weight functors. Since $\tilde {\mathfrak t} / W$ is an uncountably infinite set, is a very large algebra. Luckily, it naturally decomposes into summands: consider the partition of $\tilde {\mathfrak t} / W$ into the disjoint union of the images of orbits of the extended affine Weyl group $\widehat {W}=N_{G(\mathcal {K})}(T)/T\cong W\ltimes \mathfrak t_{{\mathbb Z}}$ on $ \tilde {\mathfrak t}$ . As discussed in [Reference WebsterWeba, Section 2.5], the space of natural transformations is nonzero if and only if $\gamma $ and $\gamma ^{\prime }$ lie in the image $\bar {\mathscr S} \subset \tilde {\mathfrak t} / W$ of a $\widehat {W}$ –orbit $\mathscr {S} \subset \tilde {\mathfrak t}$ . Thus, an indecomposable Gelfand-Tsetlin module must have weights concentrated on the image of single orbit $\mathscr {S}$ and the category of all Gelfand-Tsetlin modules decomposes as a direct sum subcategories of modules with weights concentrated on the image of single orbit.

Consider a set $\mathsf {S}\subset \tilde {\mathfrak t}$ , and its image $\bar {\mathsf {S}} \subset \tilde {\mathfrak t} / W$ . We defineFootnote ² to be the category of Gelfand-Tsetlin modules modulo the subcategory of modules killed by for all $\gamma \in \bar {\mathsf {S}}$ . In the case where $\mathsf {S}=\mathscr {S}$ is a single $\widehat {W}$ -orbit, is just the subcategory of modules with support in $\bar {\mathscr {S}}$ .

Consider the algebra . More generally, given two sets $ \mathsf S, \mathsf S^{\prime } \subset \tilde {\mathfrak t}$ , we let

(4.1)

which is an $ F(\mathsf S^{\prime }), F(\mathsf S) $ bimodule.

We recall the following definition from [Reference WebsterWeba, Definition 2.24].

By [Reference WebsterWeba, Corollary 4.16], a complete set $ \mathsf {S}$ always exists. In this case, is equal to . Moreover, if $\mathsf {S},\mathsf {S}^{\prime }$ are both complete sets for $ \mathscr {S}$ , then $F(\mathsf {S}, \mathsf {S}^{\prime }) $ gives a Morita equivalence between $ F(\mathsf {S}) $ and $ F(\mathsf {S}^{\prime }) $ . Note that the existence of a complete set shows that is Artinian, since the functor is an equivalence and any module in its image is finite-dimensional.

4.3 Category $\mathcal {O}$

The Coulomb branch $\mathcal A(G,N)$ has a Hamiltonian action of the torus $K=(G/[G,G])^\vee $ , so we have a Hamiltonian ${\mathbb C}^\times $ -action on $ \mathcal A(G,N)$ for each character $\chi $ of G.

More precisely, the map $\chi \colon G\to {\mathbb C}^\times $ induces a map $ \mathsf {Gr}_G \rightarrow \mathsf {Gr}_{{\mathbb C}^\times }$ . Since $\mathsf {Gr}_{{\mathbb C}^\times } = {\mathbb Z}$ , this gives a ${\mathbb Z}$ -grading to $ \mathcal A(G,N) $ , and thus an action of ${\mathbb C}^\times $ . By abuse of notation, we let $\chi $ also denote the derivative of the character $ \chi $ , interpreted as an element of $(\mathfrak t^*)^W$ . Then by [Reference Braverman, Finkelberg and NakajimaBFN18, Lemma 3.19], we have the quantum moment map relation: for $a \in \mathcal A(G,N)$ an element of weight $k \in {\mathbb Z}$ (with respect to $ \chi $ ), we have $[\chi ,a]=ka$ .

As discussed in [Reference WebsterWebb, Theorem 4.10], we can realize category ${{\mathcal A(G,N)}{\text {-}\mathcal {O}}}$ using the same algebraic approach that we used to understand the category . The set $\bar {\mathscr {S}}$ is divided into finitely many equivalence classes by the relation $\gamma \sim \gamma ^{\prime }$ if as functors. On each equivalence class, either:

• • Thought of as a function on $\bar {\mathscr {S}}$ , the function $\gamma \mapsto \Re (\chi (\gamma ))$ attains a maximum, and attains it at a finite number of points. In this case, we call the equivalence class bounded.

• • On $ \bar {\mathscr S}$ , the function $\gamma \mapsto \Re (\chi (\gamma ))$ either has no maximum, or it attains its maximum on an infinite set. In this case, we call the equivalence class unbounded.

Thus, let ${{\mathcal A(G,N)}{\text {-}\mathcal {O}}}_{\mathscr {S}}$ denote the subcategory of given by objects in ${{\mathcal A(G,N)}{\text {-}\mathcal {O}}}$ . For any $ \mathsf S \subset \tilde {\mathfrak t}$ , let $\mathscr {I}\subset F(\mathsf {S})$ be the two-sided ideal generated by the identity on for all $\gamma \in \bar {\mathsf {S}}$ whose equivalence class is unbounded (alternatively, this is the span of the natural transformations that factor through such ).

This gives us the following modification of Proposition 4.5:

As mentioned above, this is a rephrasing of [Reference WebsterWebb, Theorem 4.10]: in the notation of that paper, the algebra $F(\mathsf {S})$ would be denoted $A_{\mathsf {S}}$ and the ideal $\mathscr {I}$ by $\mathscr {I}_{-\chi }$ .

4.4 Gelfand-Kirillov dimension

Let $\mathbb {k}$ be a field. Consider a $\mathbb {k}$ -algebra A which is generated by a finite-dimensional subspace $A_0$ , and a left A-module M which is finitely generated by a finite-dimensional subspace $M_0$ . In this context, the Gelfand-Kirillov dimension $\operatorname {GKdim}_A(M)$ is defined by:

(4.2) $$ \begin{align} \operatorname{GKdim}_A(M)=\limsup_{n\to \infty }\log_n\dim_{\mathbb{k}}(A_0^nM_0) .\end{align} $$

It’s a standard result that this number is independent of the choice of $A_0$ and $M_0$ , and only depends on the structure of M as an A-module.

Given a Gelfand-Tsetlin module M for $\mathcal A(G,N)$ , let $M^B$ denote the corresponding module of the Morita equivalent algebra $\mathcal A^B(G,N)$ (see Theorem 2.14(1) and Section 5.1 below). Recall that $\mathcal A^B(G,N)$ contains $\mathcal A(T,N)$ as a subalgebra, by Proposition 2.5.

Proof. $(1) \geq (2)$ : This is a consequence of:

$$ \begin{align*}\operatorname{GKdim}_{\mathcal A(G,N)}(M) = \operatorname{GKdim}_{\mathcal A^B(G,N)}(M^B) \geq \operatorname{GKdim}_{\mathcal A(T,N)}(M^B). \end{align*} $$

Here, the equality on the left is due to the Morita equivalence of $\mathcal A(G,N)$ and $\mathcal A^B(G,N)$ , while the inequality on the right comes simply from the fact that $\mathcal A(T,N) \subset \mathcal A^B(G,N)$ is a subalgebra.

$(2) = (3)$ : This follows from the same argument as [Reference Musson and Van den BerghMV98, Proposition 8.2.3]. In fact, if the action of T on N is faithful, then by [Reference Braverman, Finkelberg and NakajimaBFN18, Section 4(vii)], the algebra $\mathcal A(T,N)$ is precisely one of the algebras $B^{\chi }$ considered by [Reference Musson and Van den BerghMV98].

$(3)\geq (1)$ : As noted above, we may equivalently look at the Gelfand-Kirillov dimension of $M^B$ over $\mathcal A^B(G,N)$ . Now, if $a \in \mathcal A^B(G,N)$ is any element, then there exist elements $w_1,\ldots , w_p\in \widehat {W}$ of the extended affine Weyl group, such that

(4.3)

for every weight $\lambda \in \tilde {\mathfrak t}$ . This claim follows from the results of [Reference WebsterWeba] (see also [Reference Futorny and OvsienkoFO14, Lemma 3.1(4)]). Explicitly, the localization of $\mathcal A^B(G,N)$ at the fraction field of is isomorphic to the smash product . The image $a \mapsto \sum _{i=1}^p a_i w_i$ in this localization is then a finite sum, where and $w_i \in \widehat {W}$ . These are precisely the desired elements $w_i$ .

Choose any finite-dimensional generating subspace $A_0$ of $\mathcal A^B(G,N)$ , which exists by [Reference Braverman, Finkelberg and NakajimaBFN18, Proposition 6.8]. Then we can choose $w_1,\ldots , w_p \in \widehat {W}$ , such that (4.3) holds uniformly for all $a \in A_0$ .

Now, fix any W-invariant norm on $ \tilde {\mathfrak t}$ . Note that each $w_i$ above is given by an element of the finite Weyl group times a translation $\nu _i$ . Let $\epsilon $ be the maximum of the norms of the these elements $\nu _i$ . Thus, by the triangle inequality, the ball $B(t)$ of radius t around $0$ satisfies $w_i\cdot B(t)\subset B(t+\epsilon )$ for any $t>0$ . Consequently, by (4.3), we have

Next, choose any finite-dimensional generating set $ M_0$ for $M^B$ . It is contained in a sum of finitely many weight spaces, so we can choose $ t_0$ so that . Thus, we find that

We conclude that the Gelfand-Kirillov dimension satisfies:

Consider the equivalence relation $\lambda \sim \lambda ^{\prime }$ iff . By [Reference WebsterWeba, Corollary 4.16], each $ \widehat {W}$ -orbit only contains finitely many equivalence classes, called clans, and each is the W-orbit of an intersection of a $\mathfrak t_{{\mathbb Z}}$ -coset with a convex polyhedron. Each indecomposable Gelfand-Tsetlin module is supported on a single $ \widehat {W}$ orbit (this follows from (4.3); see also the discussion before Proposition 4.5), so by finite generation, $\operatorname {Supp}(M^B)$ is contained in a finite union of $\widehat {W}$ -orbits. Thus, there are only finitely many equivalence classes $\{U_1,\dots , U_r\}$ in $\operatorname {Supp}(M^B)$ .

The Zariski closure $\overline {\operatorname {Supp}(M^B)}$ is thus the union of the Zariski closures $\overline {U_i}$ . Thus, if we define $ d= \dim \overline {\operatorname {Supp}(M^B)} $ , then $d=\max _{i=1}^r\dim (\overline {U_i})$ . Similarly, we have an equality

where for any $ \lambda \in U_i$ (these dimensions are constant by the definition of $ U_i$ ).

Since $U_i$ is a finite union of the intersections of $\mathfrak t_{{\mathbb Z}}$ -cosets with convex polyhedra, this latter growth rate is exactly the dimension of its Zariski closure. This shows that

which completes the proof that $(3) \geq (1)$ .

Finally, $(1) \geq (4)$ is a general property of passing to associated graded algebras and modules. Indeed, if the Krull dimension of $\operatorname {gr} M$ is q, then by Noether normalization, there are q algebraically independent elements $\bar a_1,\dots , \bar a_q$ of $\operatorname {gr} \mathcal A(G,N)$ , and which act on some element $\bar {v}\in \operatorname {gr} M$ with no relations beyond commutativity. Thus, if $v,a_1,\dots , a_q$ are preimages of these elements, and $A_0$ the span of $a_1,\dots , a_q$ , then we have $\dim A_0^n\cdot v\geq \binom {n+q-1}{q-1}$ , so this shows that $\operatorname {GKdim}_{\mathcal A(G,N)}(M)\geq q$ .

Let us include here a closely related result:

Lemma 4.12. Let M be a Gelfand-Tsetlin module for $\mathcal A(G,N)$ . Then

$$\begin{align*}\operatorname{GK-dim}(\mathcal A(G,N)/\operatorname{ann}(M))=2\operatorname{GK-dim}(M).\end{align*}$$

Proof. In the case $G=T$ , this proven by Musson and van der Bergh in [Reference Musson and Van den BerghMV98, Corollary 8.2.5].

Thus, in the general case, exactly as in the proof of Lemma 4.11, we can consider $M^B$ over the Morita equivalent algebra $\mathcal A^B(G,N)$ . Let $A=\mathcal A^B(G,N)/\operatorname {ann}_{\mathcal A^B(G,N)}(M^B)$ and $A_T=\mathcal A(T,N)/\operatorname {ann}_{\mathcal A(T,N)}(M^B)$ . The Abelian case proves that

$$\begin{align*}\operatorname{GK-dim}(A_T)=2\operatorname{GK-dim}(M^B),\end{align*}$$

where the latter is regarded as an $\mathcal A(T,N) $ module. By the previous Lemma, we have $ \operatorname {GK-dim}(M)=\operatorname {GK-dim}(M^B)$ and so it suffices to prove that

$$\begin{align*}\operatorname{GK-dim}(A) = \operatorname{GK-dim}(A_T).\end{align*}$$

Since $\mathcal A(T,N)/\operatorname {ann}_{\mathcal A(T,N)}(M^B)$ is a subalgebra of $\mathcal A^B(G,N)/\operatorname {ann}(M^B)$ , we thus have

$$\begin{align*}\operatorname{GK-dim}(A)\geq \operatorname{GK-dim}(A_T).\end{align*}$$

Thus, we need only prove the opposite inequality. For $w\in \widehat {W}$ , let $R_{G,N}^B(\leq w)$ be the preimage of the Schubert variety $\overline {IwI}/I$ in $R_{G,N}^B$ . Let $\mathcal A^B(\leq w)$ be the homology of this subspace; this is a -subbimodule of $\mathcal A^B(G,N)$ . Consider $\mathcal A^B(\leq w)/\mathcal A^B(< w)$ . This quotient is the homology $H_*^{\tilde {I}\rtimes {\mathbb C}^{\times }}(IwI/I)$ , which is a free module of rank 1 over as a left module or as a right module, with the two actions differing by the action of w. The image of in A is a quotient of this polynomial ring. Note that for a simple $\mathcal A(T,N)$ GT-module $M^{\prime }$ , the kernel of the map is the intersection of the maximal ideals associated to the nonzero weight spaces, and is thus the radical ideal of polynomials vanishing on $\overline {\operatorname {Supp}(M^{\prime })}$ ; for example, this follows from [Reference Musson and Van den BerghMV98, Proposition 3.1.7].

This implies that the kernel for an arbitrary $\mathcal A(T,N)$ GT-module lies in the product of finitely many such ideals. In particular, the image of the map is a not-necessarily-radical quotient, whose support as a coherent sheaf is $\overline {\operatorname {Supp}(M^B)}$ .

Thus, the same is true of any subquotient of A as a left or right -module, considered as a quasi-coherent sheaf. In particular, taking the corresponding quotient $A(\leq w)/A(<w)$ , we thus obtain a -bimodule whose support as a left and a right module must be in $\overline {\operatorname {Supp}(M^B)}$ . Since these actions differ by w, the support as a left module must lie in $\overline {\operatorname {Supp}(M^B)}\cap w\cdot \overline {\operatorname {Supp}(M^B)}$ . Now, note that all the components of $\overline {\operatorname {Supp}(M^B)}$ are affine subspaces which are $\leq d$ - dimensional, since the Zariski closure of a clan has this form; now, for any two affine subspaces $E_1,E_2$ of an n-dimensional affine space, modelled on vector subspaces $V_1,V_2$ which are $m_1$ - and $m_2$ -dimensional. The set of elements x, such that $E_1\cap E_2\neq \emptyset $ is an affine subspace itself, modelled on the span $V_1+V_2$ , and this intersection is an affine subspace modelled on $V_1\cap V_2$ . If we apply this with $E_1$ , a component of $\overline {\operatorname {Supp}(M^B)}$ , and $E_2$ , the image of such a component under an element of the finite Weyl group, we find that the intersection $\overline {\operatorname {Supp}(M^B)}\cap w\cdot \overline {\operatorname {Supp}(M^B)}$ is $\geq k$ -dimensional only for extended affine Weyl group elements whose translation parts lie in a $2d-k$ -dimensional variety, where $d=\dim \overline {\operatorname {Supp}(M^B)}$ .

Choose a generating set of the extended affine Weyl group containing the simple reflections, let $\ell _0(w)\geq \ell (w)$ be the length function of with respect to these generators (since $G $ is not usually semisimple, we will need extra generators). Let $\mathcal A^B(\leq n)$ be the span of $\mathcal A^B(\leq w)$ for all elements w with $\ell _0(w)\leq n$ . Now, consider the span $\mathcal A_0$ of

1. 1. the degree 1 elements and

2. 2. generators of $ \mathcal A^B(\leq n)$ as a left -module for some large n.

For n sufficiently large, this will be a set of generators of $\mathcal A^B(G,N)$ as an algebra. Let $A_0$ be the image of $\mathcal A_0$ in A. We need to show that the dimension of $A_0^m$ does not grow too quickly in terms of m; in order to obtain this bound, we use the filtration $A(\leq w)$ discussed above, and the loop filtration $F_p\mathcal A$ induced by the homological grading before we specialize $\hbar $ to 1.

In particular, we let D be the smallest integer, such that $\mathcal A_0\subset F_{D}\mathcal A^B$ .

Thus, we have $\mathcal A_0^m\subset \mathcal A^B(\leq nm)\cap F_{Dm}\mathcal A$ .

If $\dim \overline {\operatorname {Supp}(M^B)}\cap w\cdot \overline {\operatorname {Supp}(M^B)} \leq k$ , then we must have that the dimension of $F_pA(\leq w)/F_pA(<w)$ must be bounded by $D^{\prime \prime } p^k$ for all p for some constant $D^{\prime \prime }$ ; since this intersection is a union of affine spaces, whose number of components is bounded by the number of pairs of components, we can choose one $D^{\prime \prime }$ which works for all w. Combining our estimates, we find:

1. 1. The number of $w\in \widehat {W}$ with $\ell _0(w)\leq nm$ , such that $\dim \overline {\operatorname {Supp}(M^B)}\cap w\cdot \overline {\operatorname {Supp}(M^B)}=k$ is bounded above by $D^{\prime }m^{2d-k}$ .

2. 2. The dimension of $(A(\leq w)\cap A_0^m)/(A(< w) \cap A_0^m)$ is bounded above by $D^{\prime \prime }(Dm)^k$ if $\ell _0(w)\leq nm$ .

Thus, summing over $k=1,\dots , 2d$ , we have $\dim A_0^m\leq D^{\prime }D^{\prime \prime }D^{2d}m^{2d}$ . Thus, we have

$$\begin{align*}\log_m(\dim A_0^m)\leq 2d +\frac{\log(2dD^{\prime}D^{\prime\prime})D^{2d}}{\log m} \end{align*}$$

so taking the limit, we have $\operatorname {GK-dim}(A)\leq 2d$ , completing the proof.

5.1 Restriction and induction functors

The relations of Theorem 2.14 induce a number of functors.

1. 1. The Morita equivalence of Proposition 2.4 gives us an equivalence of categories

   $$ \begin{align*} \mathcal A(G,N)\operatorname{-mod} \cong \mathcal A^P(G,N)\operatorname{-mod}. \end{align*} $$

   This functor $ \mathcal A^P(G,N)\operatorname {-mod} \rightarrow \mathcal A(G,N)\operatorname {-mod} $ can be written as $ M \mapsto e^{\prime }M $ . Let $M^P=\mathcal A^P e^{\prime }\otimes _{\mathcal A}M$ be the inverse equivalence.

2. 2. Restriction induces a functor

   $$ \begin{align*} \mathcal A^P(G,N)\operatorname{-mod} \to \mathcal A(L,N) \operatorname{-mod}. \end{align*} $$

   This functor has a left adjoint given by $\mathcal A^P(G,N)\otimes _{\mathcal A(L,N)}-$ and a right adjoint $\operatorname {Hom}_{\mathcal A(L,N)}(\mathcal A^P(G,N), -)$ .

3. 3. Tensor product $\mathcal A(L, N^{\xi }_0)\otimes _{\mathcal A(L, N)} -$ induces an exact functor

   $$ \begin{align*} \mathcal A(L,N)\operatorname{-mod} \to \mathcal A(L, N^{\xi}_0)\operatorname{-mod}. \end{align*} $$

   This has a right adjoint given by restriction.

Definition 5.1. We define the functor $ \operatorname {res}_\xi =\operatorname {res} \colon \mathcal A(G,N) \operatorname {-mod}\to \mathcal A(L, N^{\xi }_0)\operatorname {-mod}$ as the composition of the above three functors

$$ \begin{align*} \mathcal A(G,N) \operatorname{-mod} \rightarrow \mathcal A^P(G,N) \operatorname{-mod} \rightarrow \mathcal A(L,N) \operatorname{-mod} \rightarrow \mathcal A(L, N^\xi_0) \operatorname{-mod}. \end{align*} $$

5.2 Functors on Gelfand-Tsetlin modules

Let us examine a little more carefully the effect of $\operatorname {res}$ on weight spaces.

5.2.1 Morita equivalence on GT weight spaces

The effect of the Morita equivalence from Proposition 2.4 on weight spaces is covered in [Reference WebsterWeba, Lemma 2.8]. Let $\nu \in \tilde {\mathfrak t}/W_L$ , and $\gamma $ its image in $ \tilde {\mathfrak t}/W$ . The preimage of $\gamma $ in $\tilde {\mathfrak t}$ is a single orbit of the Weyl group W, and $\nu $ corresponds to an orbit of $W_L$ within this larger orbit. Generically, both these orbits are free, but there are degenerate cases where they are not. The effect of the Morita equivalence on weight spaces is a bit subtle in the latter case. Let $\lambda \in \tilde {\mathfrak t}$ be a preimage of $\nu $ , let $W^{\lambda }$ be its stabilizer in W and $W_L^{\lambda }$ its stabilizer in $W_L$ .

Lemma 5.4. Let $ \lambda , \nu , \gamma $ correspond as above. Let $ M $ be an $\mathcal A(G,N)$ -module, and $M^P$ the corresponding $\mathcal A^P(G,N)$ -module. If $W^\lambda = W_L^\lambda $ , then ; more generally, we have a functorial isomorphism , where $ k = [W^\lambda : W^{\lambda }_L]$ .

Proof. Let $e^{\prime }({\lambda })$ be the symmetrizing idempotent for $W^\lambda $ and $e^{\prime }_L({\lambda })$ be the symmetrizing idempotent for $W^\lambda _L$ . Let $M^B$ be the corresponding module over $\mathcal A^B$ . By [Reference WebsterWeba, Lemma 3.2], we have that

and is a free module over ${\mathbb C}[W^\lambda ]$ . Since $e^{\prime }_L({\lambda }){\mathbb C}[W^{\lambda }]\cong {\mathbb C}^k$ , the result follows.

5.2.3 Inverting $ r_\xi $ and weight spaces

Finally, we wish to consider the effect of inverting $r_\xi $ . This can be computed separately on the summands of any decomposition into ${\mathbb C}[r_{\xi }]$ -submodules. A Gelfand-Tsetlin module has a natural such decomposition, given by the sum of weight spaces in a single ${\mathbb Z}\xi $ -coset.

Let $ \nu \in \tilde {\mathfrak t}/W_L$ . Note that $ \nu + \xi $ is well-defined since $ \xi $ is invariant under the action of $ W_L $ . Let $M $ be a $ \mathcal A(L,N)$ module. We consider the directed system

(5.1)

Let denote the direct limit of this system (here, $[\nu ]$ denotes the image of $ \nu $ in $ \left ( \tilde {\mathfrak t} / W_L \right ) / {\mathbb Z}\xi $ ).

Lemma 5.5. Let $ M $ be a Gelfand-Tsetlin $\mathcal A(L,N)$ -module. Then $ \mathcal A(L,N^{\xi }_0) \otimes _{\mathcal A(L,N)} M $ is also a Gelfand-Tsetlin module, and for any $ \nu \in \tilde {\mathfrak t}/W_L $ , we have

In particular, this functor is exact on the category of Gelfand-Tsetlin modules.

Proof. Let $M^{\prime }= \mathcal A(L,N^{\xi }_0) \otimes _{\mathcal A(L,N)} M $ . We have an obvious map $p\colon M\to M^{\prime }$ which leads to maps , which are compatible with the directed system (5.1). This induces a map .

Since $ \mathcal A(L,N^\xi _0) = \mathcal A(L,N)[r_\xi ^{-1}]$ , for any , we have that $r_\xi ^{k}v$ is in the image of M for k sufficiently large. This shows that the map is surjective.

On the other hand, if there is an element of the kernel, it is represented by some for some k. Since this element is killed by p, we have that $r_{\xi }^{k^{\prime }}w=0$ for some $k^{\prime }$ . Thus, w has trivial image in the directed limit. This shows injectivity.

However, the limit is already isomorphic to a weight space of M but possibly for a different $\nu $ . To see this, recall that in $ \mathcal A(L,N)$ , we have by (2.4) that

(5.2) $$ \begin{align} \nonumber r_{-\xi} r_\xi & = \left( \prod_{\langle\mu,\xi \rangle>0} \prod_{j = 1}^{\langle \mu, \xi \rangle}( \mu - j) \right) \left(\prod_{\langle\mu,\xi \rangle <0} \prod_{j = 0}^{-\langle \mu, \xi \rangle-1}( \mu + j )\right) \\ r_{\xi} r_{-\xi} & = \left( \prod_{\langle\mu,\xi \rangle <0} \prod_{j = 1}^{-\langle \mu, \xi \rangle}( \mu - j ) \right) \left(\prod_{\langle\mu,\xi \rangle>0} \prod_{j = 0}^{\langle \mu, \xi \rangle-1}( \mu + j )\right), \end{align} $$

where the products both range over subsets of the weights $ \mu $ of the representation $ N $ , counted with multiplicity.

These formulas motivate the following definition.

Definition 5.6. $\lambda \in \tilde {\mathfrak t} $ is called $\xi $ -negative, if

1. 1. there is no weight $ \mu $ of $ N $ , such that $ \langle \mu , \xi \rangle>0 $ and $ \langle \mu , \lambda \rangle $ is a positive integer and

2. 2. there is no weight $ \mu $ of $ N $ , such that $ \langle \mu , \xi \rangle <0 $ and $ \langle \mu , \lambda \rangle $ is a nonpositive integer.

3. 3. The stabilizer $ W^\lambda $ lies in $W_L$ , that is $W^{\lambda }=W_L^\lambda $ .

Since $ \xi $ is invariant under $ W_L $ and the set of weights of $ N $ is invariant under $ W_L $ , the set of $\xi $ -negative elements of $ \tilde {\mathfrak t} $ is invariant under $W_L $ . Thus, it makes sense to speak of $\xi $ -negative elements of $ \tilde {\mathfrak t} / W_L $ .

Lemma 5.7. Assume that $\nu \in \tilde {\mathfrak t} / W_L $ is $\xi $ -negative.

1. 1. For all $ k \in {\mathbb Z}_{\geq 0} $ , $r_\xi $ gives an isomorphism of functors

   

2. 2. The natural map is an isomorphism of functors.

Proof. Let $ M $ be a Gelfand-Tsetlin $\mathcal A(L,N)$ -module. Let $ k \in {\mathbb Z}_{\ge 0} $ . We consider the composition $ r_{-\xi } r_\xi $ as a linear operator on . By (5.2), the eigenvalues of this operator on this weight space are given by the set

$$ \begin{align*} \big\{ \langle \mu, \nu \rangle - k\langle \mu, \xi \rangle - j \ :\ \langle\mu, \xi\rangle> 0,\ 1 \leq j \leq \langle\mu,\xi\rangle \big\} \end{align*} $$

$$ \begin{align*} \cup\ \big\{ \langle \mu, \nu \rangle - k\langle \mu, \xi \rangle +j \ : \ \langle\mu, \xi\rangle < 0,\ 0 \leq j < - \langle\mu,\xi\rangle\big\}. \end{align*} $$

The hypothesis of $ \xi $ -negativity ensures that none of these eigenvalues vanish. Thus, $ r_{-\xi } r_\xi $ is an isomorphism. Similarly, we see that $ r_\xi r_{-\xi } $ is an isomorphism on .

So we conclude that $ r_\xi $ is an isomorphism. Then the second part follows immediately.

5.2.4 Combined effect of $res$ on weight spaces

Combining the results above, we conclude the following.

Theorem 5.8. Consider a Gelfand-Tsetlin $\mathcal A(G,N)$ module M. Let $ \nu \in \tilde {\mathfrak t}/W_L$ , and let $ \gamma \in \tilde {\mathfrak t}/W $ denote the image of $ \nu $ . Assume that $ \nu $ is $\xi $ -negative. Then there is a natural isomorphism

Corollary 5.9. Let $ \nu , \nu ^{\prime }\in \tilde {\mathfrak t}/W_L$ be $ \xi $ -negative, and let $ \gamma , \gamma ^{\prime } \in \tilde {\mathfrak t}/W $ be their images. There is a morphism

Proof. By Theorem 5.8, we have

Given any , the map gives us our desired map .

5.3 Algebraic description of the res functor

We will now combine Proposition 4.5 with Corollary 5.9 to obtain an algebraic description of the functor res.

Consider a $\widehat {W}$ -orbit $\mathscr {S}\subset \tilde {\mathfrak t}$ . As before, we let be the category of Gelfand-Tsetlin modules supported on the image of this orbit in $ \tilde {\mathfrak t}/W$ . Since $\mathscr {S}$ is closed under addition by $\xi $ , Lemma 5.5 shows that if M is supported on $\bar {\mathscr {S}}$ , then $\operatorname {res}(M)$ is supported on $\bar {\mathscr {S}}$ as well.

The set $\mathscr {S}$ is a finite union of $ \widehat {W}^L$ -orbits. We will fix attention on a single one of these $\widehat {W}^L$ -orbits, which we denote $\mathscr {S}^L$ . Let be the functor given by applying $\operatorname {res}$ and then taking the summand supported on $\bar {\mathscr {S}^L}$ .

Let $\mathsf {S},\mathsf {S}^L$ be finite complete sets in $\mathscr {S}, \mathscr {S}^L$ , respectively, and without loss of generality, assume that $\mathsf {S}^L\subset \mathsf {S}$ .

Since $ r_\xi $ is invertible in $ \mathcal A(L, N^\xi _0) $ as in Theorem 2.9, we have for any $ \nu \in \tilde {\mathfrak t}/W_L $ . Thus, without loss of generality, we can choose all elements of $\mathsf {S}^L$ to be $\xi $ -negative.

By Proposition 4.5, we have equivalences

We can define natural $F^L(\mathsf {S}^L), F(\mathsf {S})$ -bimodules corresponding to the restriction and induction functors:

(5.3)

where $ \bar {\mathsf S} $ denotes the image of $ \mathsf S $ in $\tilde {\mathfrak t} / W $ and $ \bar {\mathsf S^L} $ denotes the image of $ \mathsf S^L $ in $\tilde {\mathfrak t} / W_L $ .

Recall that by construction, each $\nu \in \mathsf {S}^L$ is $\xi $ -negative. Thus, by Theorem 5.8, we can choose an isomorphism , where $\gamma $ is the image of $\nu $ in $\tilde {\mathfrak t}/W$ . This induces a vector space isomorphism $F(\mathsf S, \mathsf S^L) \cong I(\mathsf S, \mathsf S^L )$ (see (4.1) for the definition of $F(\mathsf S, \mathsf S^L)$ ). The right action of $F(\mathsf S^L)$ on $ F(\mathsf S, \mathsf S^L) $ and the right action of $F^L(\mathsf S^L) $ on $I(\mathsf S, \mathsf S^L )$ are related by the homomorphism of Corollary 5.9.

Let $e^L\in F(\mathsf {S})$ be the idempotent obtained by summing the identities on elements of $\mathsf {S}^L$ . In this case, $e^LF(\mathsf {S}) e^L=F(\mathsf {S}^L)$ and $I(\mathsf {S}, \mathsf {S}^L)=e^LF(\mathsf {S}) $ . Note that for $ M \in F(\mathsf S)\operatorname {-mod}$ , we have functorial isomorphisms

(5.4) $$ \begin{align} I(\mathsf{S}, \mathsf{S}^L) \otimes_{F(\mathsf S)} M\cong e^LM\cong \operatorname{Hom}_{F(\mathsf{S})}(I(\mathsf{S}^L, \mathsf{S}),M). \end{align} $$

Theorem 5.10. We have a commutative diagram

Note that (5.4) allows us to construct left and right adjoints to $\operatorname {res}$ on the category of GT modules:

The right adjoint $\operatorname {coind}$ is well-defined on all modules, based on the definition in Section 5.1. Note that it’s not clear that $\operatorname {ind}$ is well-defined for all modules, though it seems likely that it agrees with $\operatorname {coind}$ for the coweight $-\xi $ . We can also construct $\operatorname {ind}$ from $\operatorname {coind}$ by using duality on the category of Gelfand-Tsetlin modules.

5.4 Hamiltonian reduction

In this section, we consider Part (4) of Theorem 2.14, and the corresponding functors on modules induced by quantum Hamiltonian reduction.

Let A be an algebra and $b\in A$ , and recall that $A \mathbin {/\mkern -6mu/}_1 b = \operatorname {End}_A( A / (b-1) A)$ . There is a natural $A \mathbin {/\mkern -6mu/}_1 b, A$ –bimodule structure on $A / (b-1) A$ . In particular, there is a natural functor $A\operatorname {-mod} \rightarrow A\mathbin {/\mkern -6mu/}_1 b \operatorname {-mod}$ defined by

$$ \begin{align*}M\ \mapsto \ A / (b-1) A \otimes_A M \cong M / (b-1) M, \end{align*} $$

which has right adjoint $\operatorname {Hom}_{A \mathbin {/\mkern -6mu/}_1 b}( A / (b-1) A, -)$ .

An instructive example to think about is when $G={\mathbb C}^\times $ and the action on N is trivial (in this case, the algebra $\mathcal A({\mathbb C}^\times ,N)$ does not depend on N). By [Reference Braverman, Finkelberg and NakajimaBFN18, Section 4(iv)], the resulting algebra is generated by $x=r_{-1},x^{-1}=r_{1}$ and a with the relation $[a,x]=x$ , where $ a $ is the equivariant parameter coming from . We can either think of this algebra as a ring of difference operators on the polynomial ring ${\mathbb C}[a]$ , with x acting by translation, or as differential operators on the torus ${\mathbb C}^\times $ with coordinate x, with $a=x\frac {\partial }{\partial x}$ (the isomorphism relating these two realizations is called the Mellin transform). In this case, the quantum Hamiltonian reduction is simply $\mathcal A({\mathbb C}^\times , N) \mathbin {/\mkern -6mu/}_1 x \cong {\mathbb C}$ .

The representation theory of $\mathcal A({\mathbb C}^\times ,N)$ is not trivial, but it is simple. A Gelfand-Tsetlin module M over $\mathcal A({\mathbb C}^\times ,N)$ is one on which a acts locally finitely, and can equivalently be thought of as a D-module on ${\mathbb C}^\times $ on which $a=x\frac {\partial }{\partial x}$ acts locally finitely; this implies that M corresponds to a regular local system. Thus, a simple module of this type must be a 1-dimensional local system with monodromy $c \in {\mathbb C}^\times $ . This module is isomorphic to $ \mathcal A / \mathcal A(a-m) $ for any m, such that $ e^{2\pi i m} = c $ .

For a general module $ M$ , $M/(x-1)M$ is the fibre of the local system at $x=1$ . In particular, for a Gelfand-Tsetlin module, the number of simple composition factors is the dimension of this fibre. In order to reconstruct M, we need to also remember the action of the monodromy map $\exp (2 \pi i a)$ on this quotient (which is well-defined).

Now, consider the situation of Theorem 2.11: Assume that ${\mathbb C}^{\times }_{\xi }\subset G$ is the image of a primitive cocharacter $\xi \colon {\mathbb C}^{\times }\to G$ which acts trivially on a representation N. Recall from Theorem 2.11 that, in this case, $\mathcal A(G,N) \mathbin {/\mkern -6mu/}_1 r_\xi \ \cong \ \mathcal A(G/{\mathbb C}^\times _\xi , N)$ .

Let $\tilde {\mathfrak t}^{\prime }=\tilde {\mathfrak t}/{\mathbb C} \xi $ , where we use $\xi $ to denote the derivative of the cocharacter $\xi $ . That is, $\tilde {\mathfrak t}^{\prime }$ is the Lie algebra of a maximal torus of $\tilde {G}^{\prime }=\tilde G/{\mathbb C}^{\times }_{\xi }$ . Thus, we have a map ${\mathbb C}[ \tilde {\mathfrak t}^{\prime }]^W\subset {\mathbb C}[\tilde {\mathfrak t}]^W\to \mathcal A(G,N)$ ; the composed map commutes with $r_{\xi }$ , so ${\mathbb C}[\tilde {\mathfrak t}^{\prime }]^W$ maps into the Hamiltonian reduction. This induces the usual map .

We have a Hamiltonian reduction functor on left modules:

$$ \begin{align*}M \mapsto M/(r_{\xi}-1)M.\end{align*} $$

Proposition 5.11. Let . Then $ M / (r_\xi - 1) M $ is a Gelfand-Tsetlin module for $\mathcal A(G^{\prime }, N) $ , and for $ \gamma ^{\prime } \in \tilde {\mathfrak t}^{\prime } /W$ we have

where the direct sum ranges over a set of representatives modulo $ {\mathbb Z}\xi $ for the preimage of $ \gamma ^{\prime } $ in $ \tilde {\mathfrak t}/W $ .

For $\gamma $ a ${\mathbb Z}\xi $ -coset representative, the weight space can be written more canonically as the limit of the directed system (5.1). Since ${\mathbb C}^{\times }_{\xi }$ acts trivially on N, every map of this directed system will be an isomorphism.

Proof. For any $\gamma \in \mathfrak t/W$ , the weight spaces and are canonically isomorphic by $r_{\pm k\xi }$ . Since all weight spaces of M are finite-dimensional, it follows that is a free module of finite rank over the Laurent polynomial ring ${\mathbb C}[r_{\xi },r_{-\xi }]$ , which is freely generated by . Since M is finitely generated, the support of M is a finite union of orbits of the extended affine Weyl group of G, and so there are only finitely many cosets of ${\mathbb Z}\xi $ in any given coset of ${\mathbb C}\xi $ that lie in the support of M.

Given $\gamma ^{\prime } \in \tilde {\mathfrak t}^{\prime } / W$ , we have that is also a free module of finite rank over ${\mathbb C}[r_{\xi },r_{-\xi }]$ : there are only finitely many cosets $[\gamma ] \in (\gamma ^{\prime }+{\mathbb C}\xi )/{\mathbb Z}\xi $ for which , and we may apply the above argument on each coset. We can choose free generators for $M^{\prime }_{\gamma ^{\prime }}$ by fixing one representative $\gamma $ from each such coset $[\gamma ]$ , and thus

Since $\xi $ is primitive, we can find $ a \in \tilde {\mathfrak t}^*_{{\mathbb Z}} $ with $ \xi (a) = 1 $ . Choose a logarithm map $ \log : {\mathbb C}^\times \rightarrow {\mathbb C} $ inverse to $ m \mapsto e^{2 \pi i m} $ . For simplicity, we can uniquely fix this by requiring its image to lie in $[0,1)+i\mathbb R$ .

The adjoint action of a on $\mathcal A(G,N)$ integrates to a $ {\mathbb C}^\times $ action and thus has only integer eigenvalues. Thus, the module M is the direct sum of submodules

(5.5)

where denotes the generalized $ x$ -eigenspace for the action of $ a $ on $ M $ , and where $ c $ ranges over $ {\mathbb C}^\times $ .

We use $ a $ to split $ \tilde {\mathfrak t} = \tilde {\mathfrak t}^{\prime } \oplus {\mathbb C} \xi $ . For $ c \in {\mathbb C}^\times $ , define

Via Proposition 5.11, we can identify $ (M/(r_\xi - 1)M))_c $ with a subspace of $ M/(r_\xi -1)M $ .

This gives a decomposition

(5.6) $$ \begin{align} M/(r_\xi-1)M = \bigoplus_{c \in {\mathbb C}^\times} (M/(r_\xi - 1)M))_c =\bigoplus_{c \in {\mathbb C}^\times}M_c/(r_{\xi}-1)M_c \end{align} $$

into $ \mathcal A(G^{\prime },N)$ -submodules by reducing the decomposition (5.5).

To connect this to the D-modules on the punctured line described above, note that $ a, r_\xi ^{\pm } $ generate a copy of $ D({\mathbb C}^\times ) $ inside of $ \mathcal A(G,N) $ . The module $ M $ can be viewed as a D-module on $ {\mathbb C}^\times $ via this isomorphism. Then the left-hand side of (5.6) is the fibre of this D-module at $ 1 $ and the right-hand side is the decomposition of the fibre (or equivalently, the nearby cycles at the origin) into generalized eigenspaces for the monodromy around the origin.

6.1 Recollection on earlier results

By Proposition 4.5, spaces of natural transformations between weight functors control the category of Gelfand-Tsetlin modules. In the papers [Reference WebsterWeba, Reference WebsterWebb], the third author gave a geometric description of these spaces. We will now recall this description; it will be phrased as an equivalence of categories. We begin by defining the two categories involved.

In this section, we will only work with integral weights. Here, $ \tilde {\mathfrak t}_{\mathbb Z} = \operatorname {Hom}({\mathbb C}^\times , \tilde T) \subset \tilde {\mathfrak t} $ .

6.1.1 Category of weight functors

Definition 6.1. We call a Gelfand-Tsetlin module over $\mathcal A(G,N)$ or $\mathcal A^B(G,N)$ integral if its support lies in $\tilde {\mathfrak t}_{\mathbb Z}$ .

Let $ \widehat {\mathscr A}_{\mathbb Z}(G,N) $ be the category with objects $ \tilde {\mathfrak t}_{\mathbb Z} / W$ and morphisms given by

Similarly, let $\widehat {\mathscr A}_{\mathbb Z}^B(G,N) $ be the category with objects $ \tilde {\mathfrak t}_{\mathbb Z} $ and morphisms given by

where, as before, is a weight functor on the category of $ \mathcal A^B(G,N)$ -modules.

Remark 6.2. The reader might naturally wonder about the relationship between $\widehat {\mathscr A}_{\mathbb Z}(G,N) $ and $\widehat {\mathscr A}_{\mathbb Z}^B(G,N)$ . They are not equivalent: for example, in the pure case where $N=0$ , if $\gamma $ is a W-orbit with a single element, the endomorphism algebra of $\gamma $ in $\widehat {\mathscr A}(G,N) $ has a single 1-dimensional discrete irreducible representation, and no object in $\widehat {\mathscr A}^B(G,N) $ has this property. On the other hand, by Lemma 5.4, there is an equivalence between the Karoubian envelopes of these categories, sending $\lambda \in \tilde {\mathfrak t}_{\mathbb Z} $ to the direct sum of $W^\lambda $ copies of its image in $\tilde {\mathfrak t}_{\mathbb Z}/ W$ . The inverse functor thus sends an element of $\tilde {\mathfrak t}/W$ corresponding to a nonfree orbit to the image of the symmetrizing idempotent $e^{\prime }(\lambda )$ in $W^\lambda $ acting on a preimage $\lambda $ .

6.1.2 Steinberg category

Next, we will define certain Steinberg-type varieties, which will be the building blocks of our second category.

First, given coweights $ \gamma , \gamma ^{\prime } \in \tilde {\mathfrak t}_{\mathbb Z} /W $ , as before, let $ \lambda , \lambda ^{\prime } $ be the antidominant lifts of $ \gamma , \gamma ^{\prime } $ . As above, $N^{\lambda }_{\leq }$ is the subspace in N on which $\lambda $ has nonpositive weight, and let $P_\lambda \subset G$ be the parabolic subgroup on whose Lie algebra $\lambda $ has nonpositive weight (and similarly for $\lambda ^{\prime }$ ). Let:

(6.1) $$ \begin{align} Y_\gamma&=(G\times N^{\lambda}_{\leq})/{P}_\lambda\cong \{(gP_{\lambda},n) \mid n\in gN^{\lambda}_{\leq}\} \end{align} $$

(6.2) $$ \begin{align} {}_{\gamma}{\mathbb{Y}}_{{\gamma^{\prime}}}&=Y_\gamma\times_{N} Y_{\gamma^{\prime}}=\{(g_1 P_\lambda,g_2P_{\lambda^{\prime}}, n) \mid n\in g_1N^{\lambda}_{\leq}\cap g_2N_{\leq}^{\lambda^{\prime}}\}. \end{align} $$

Let $\widehat {H}^{G}( {}_{\gamma }{\mathbb {Y}}_{{\gamma ^{\prime }}})$ denote the completion of the equivariant Borel-Moore homology of $ {}_{\gamma }{\mathbb {Y}}_{{\gamma ^{\prime }}}$ with respect to its grading.

We will also need a full flag version of this construction. Let:

(6.3) $$ \begin{align} X_{\lambda} =(G\times N^{\lambda}_{\leq})/{B} \qquad {}_{\lambda}{\mathbb{X}}_{{\lambda^{\prime}}}=X_{\lambda}\times_{N} X_{\lambda^{\prime}}=\{(g_1 B,g_2 B, n) \mid n\in g_1N^{\lambda}_{\leq}\cap g_2N_{\leq}^{\lambda^{\prime}}\}. \end{align} $$

Note that $ B \subset P_\lambda $ and if $ \lambda $ is the antidominant lift of $ \gamma $ , we have a natural morphism $ X_\lambda \rightarrow Y_\gamma $ which is a $ P_\lambda / B $ bundle.

We can easily extend the definition of these spaces when $\lambda , \lambda ^{\prime }$ are not antidominant. To this end, we define $ B_\lambda $ to be the Borel subgroup whose Lie algebra consists of those root spaces $ \mathfrak {g}_\alpha $ , for all $ \alpha $ , such that either $ \langle \lambda , \alpha \rangle < 0 $ , or $ \langle \lambda , \alpha \rangle = 0 $ and $ \alpha $ is negative. Then $N^{\lambda }_{\leq }$ will be invariant under $B_{\lambda }$ and we define $X_\lambda =(G\times N^{\lambda }_{\leq })/{B_{\lambda }}$ . Note that this space would be the same for any other Borel contained in $P_{\lambda }$ ; any two such Borels are conjugate in $P_\lambda $ , and any element in $P_{\lambda }$ conjugating between them induces an isomorphism by $(g,n)\mapsto (gp^{-1},pn)$ . More generally, this assignment of spaces to coweights is equivariant for the action of the Weyl group. Even though $wB_{\lambda }w^{-1}\neq B_{w\lambda }$ in some cases, we still have $wB_{\lambda }w^{-1}\subset P_{w\lambda }$ , and so $X_{\lambda }\cong X_{w\lambda }$ . Finally, we extend the definition (6.3) of ${}_{\lambda }{\mathbb {X}}_{{\lambda ^{\prime }}}$ to the case of arbitrary $ \lambda , \lambda ^{\prime } \in \tilde {\mathfrak t}_{\mathbb Z} $ .

Let $\widehat {H}^{G}( {}_{\lambda }{\mathbb {X}}_{{\lambda ^{\prime }}})$ denote the completion of the equivariant Borel-Moore homology of $ {}_{\lambda }{\mathbb {X}}_{{\lambda ^{\prime }}}$ with respect to its grading.

Remark 6.3. When there is an ambiguity, we will write $ Y_\gamma ^{G,N}$ etc. to keep track of the group $ G $ and representation $ N $ .

Definition 6.4. Let $ \widehat {\mathscr X}(G,N) $ be the category with objects $ \tilde {\mathfrak t}_{\mathbb Z}/W $ and morphisms

$$ \begin{align*}\operatorname{Hom}_{\widehat{\mathscr X}(G,N)}(\gamma, \gamma^{\prime}) = \widehat{H}^{G}( {}_{\gamma}{\mathbb{Y}}_{{\gamma^{\prime}}}).\end{align*} $$

Similarly, let $ \widehat {\mathscr X}^B(G,N) $ be the category with objects $ \tilde {\mathfrak t}_{\mathbb Z} $ and morphisms

$$ \begin{align*}\operatorname{Hom}_{\widehat{\mathscr X}^B(G,N)}(\lambda, \lambda^{\prime}) = \widehat{H}^{G}( {}_{\lambda}{\mathbb{X}}_{{\lambda^{\prime}}}).\end{align*} $$

Composition in these categories is defined by convolution, as defined in [Reference Chriss and GinzburgCG97, (2.7.9)], viewing $ {}_{\lambda }{\mathbb {X}}_{{\lambda ^{\prime }}}\subset X_{\lambda }\times X_{\lambda ^{\prime }}$ (in the notation of [Reference Chriss and GinzburgCG97], $M_1=X_{\lambda }, M_2=X_{\lambda ^{\prime }}$ ). The associativity of the convolution product is given by [Reference Chriss and GinzburgCG97, Section 2.7.18].

The category $ \widehat {\mathscr X}^B(G,N) $ is a variation of the Steinberg category, defined in [Reference WebsterWebb, Section 2.4]. The definition from [Reference WebsterWebb] involves assigning a subspace in N to each element of $ \tilde {\mathfrak t}_{\mathbb Z} $ , which we have done here by the assignment $\lambda \mapsto N^{\lambda }_{\leq }$ ; in [Reference WebsterWebb, Section 2.4], this subspace is encoded as a sign sequence on a basis of N representing which basis vectors lie in $ N^{\lambda }_{\leq }$ and which do not.

6.1.3 The equivalences

Following [Reference WebsterWebb, Definition 4.2], we will now construct equivalences between the weight functor categories and the Steinberg categories. Our strategy will be to begin with $ \widehat {\mathscr X}(T,N), \ \widehat {\mathscr A}_{\mathbb Z}(T,N)$ , then study $ \widehat {\mathscr X}^B(G,N),\ \widehat {\mathscr A}_{\mathbb Z}^B(G,N)$ and finally $\widehat {\mathscr X}(G,N), \ \widehat {\mathscr A}_{\mathbb Z}(G,N)$ .

Recall from Section 2.3, that the algebra $\mathcal A_{\varphi }(T,N)$ is generated over by the elements $r_{\nu }$ (for $ \nu \in \mathfrak t_{\mathbb Z}$ ), with relations given by (2.4).

For $ \lambda , \lambda ^{\prime } \in \mathfrak t_{\mathbb Z}$ , we define by the following formula, where the product ranges over the weights $ \mu $ of the representation $ N $ , counted with multiplicity:

(6.4) $$ \begin{align} \Phi_0(\lambda,\lambda^{\prime}) = \prod_\mu \prod_{\substack{j = 1,\dots, -\langle \mu, \lambda-\lambda^{\prime}\rangle\\j\neq\langle\mu,\lambda^{\prime} \rangle }}( \mu - j ). \end{align} $$

Note that the polynomial $\Phi _0(\lambda ,\lambda ^{\prime })$ acts invertibly on the functor , so we can define morphisms in $\widehat {\mathscr A}_{\mathbb Z}(T,N)$ by

Note that $ {}_{\lambda ^{\prime }}{\mathbb {X}}^{T,N}_{{\lambda }} $ is a vector space. We also let $\mathbb {w}(\lambda ,\lambda ^{\prime })$ denote its fundamental class $[{}_{\lambda ^{\prime }}{\mathbb {X}}^{T,N}_{{\lambda }}] $ , which is a morphism in $\widehat {\mathscr X}(T,N)$ .

A special case of [Reference WebsterWebb, Theorem 4.3] is that:

Theorem 6.5. There is an equivalence $ \mathsf E : \widehat {\mathscr X}(T,N)\cong \widehat {\mathscr A}_{\mathbb Z}(T,N)$ which is the identity on objects. On morphisms, this functor sends an element of to the nilpotent part of the action of the same element in $\widehat {\mathscr A}_{\mathbb Z}(T,N)$ , and takes $ \mathbb {w}(\lambda ,\lambda ^{\prime }) $ to the same-named morphism.

From Proposition 2.5, we have an inclusion $ \mathcal A_{\varphi }(T,N) \rightarrow \mathcal A_{\varphi }^B(G,N)$ . This leads to a functor $ \widehat {\mathscr A}_{\mathbb Z}(T,N)\to \widehat {\mathscr A}_{\mathbb Z}^B(G,N)$ which is the identity on objects. Describing the additional generators needed to generate $\mathcal A_{\varphi }^B(G,N)$ is tricky when working purely with the algebra $\mathcal A_{\varphi }$ ; this process is simplified by considering the extended category introduced in [Reference WebsterWebb, Section 3]. Here, we proceed a little differently.

Let ${\mathbb {L}}$ denote the fraction field of and $\widehat {{\mathbb {L}}}$ the fraction field of the completion . Recall from [Reference WebsterWeba, Proposition 4.2] that $\mathcal A_{\varphi }(T,N)$ is a principal Galois order inside the skew group algebra ${\mathbb {L}}\rtimes \mathfrak t_{{\mathbb Z}}$ for the usual action of $\mathfrak t_{{\mathbb Z}}$ on ${\mathbb {L}}$ by translations on $\mathfrak t$ . Similarly, $\mathcal A_{\varphi }(G,N)$ is a principal Galois order inside the Weyl invariants of $({\mathbb {L}}\rtimes \mathfrak t_{{\mathbb Z}})^W$ . The algebra $\mathcal A_{\varphi }^B(G,N)$ is the corresponding flag order in the skew group algebra ${\mathbb {L}}\rtimes \widehat {W}$ . That is, in particular, these inclusions induce isomorphisms:

By [Reference WebsterWeba, Lemma 2.11(4)], the Hom space

$$\begin{align*}\operatorname{Hom}_{\widehat{\mathscr A}_{\mathbb Z}^B(G,N)}(\nu, \nu^{\prime})=\mathcal A_{\varphi}^B/(\mathcal A_{\varphi}^B\mathfrak{m}_{\nu}^N+\mathfrak{m}_{\nu^{\prime}}^N \mathcal A_{\varphi}^B),\end{align*}$$

is isomorphic to

(6.5)

Applying the same result with $G=B=T$ and the product decomposition $\widehat {W}=\mathfrak t_{{\mathbb Z}}\cdot W$ , we obtain an isomorphism:

(6.6)

On the other hand, we also have a functor $\widehat {\mathscr X}(T,N)\to \widehat {\mathscr X}^B(G,N)$ , which is the identity on objects and which acts on morphisms by

(6.7) $$ \begin{align} \widehat{H}^{T}( {}_{\lambda}{\mathbb{X}}^{T,N}_{{\lambda^{\prime}}}) \rightarrow \widehat{H}^{T}( {}_{\lambda}{\mathbb{X}}^{G,N}_{{\lambda^{\prime}}}) = \widehat{H}^{B}( {}_{\lambda}{\mathbb{X}}^{G,N}_{{\lambda^{\prime}}})\rightarrow \widehat{H}^{G}( {}_{\lambda}{\mathbb{X}}^{G,N}_{{\lambda^{\prime}}}), \end{align} $$

the composition of pushforward in T-equivariant homology, followed by G-saturation, the map $\widehat {H}^{B}(X)\cong \widehat {H}^{G}((G\times X)/B)\to \widehat {H}^{G}(X)$ which ‘averages’ B-equivariant cycles under G.

We can write

$$ \begin{align*} \operatorname{Hom}_{\widehat{\mathscr X}_{\mathbb Z}^B(G,N)}(\nu, \nu^{\prime})&\cong \widehat{H}^{B}(\{(g B, n) \mid n\in N^{\lambda}_{\leq}\cap gN_{\leq}^{\lambda^{\prime}}\} )\\ &\cong \widehat{H}^{T}(\{(g B, n) \mid n\in N^{\lambda}_{\leq}\cap gN_{\leq}^{\lambda^{\prime}}\} ). \end{align*} $$

The fixed points of T on the space are given by $\{(wB,n)\mid w\in W, n\in N^T \}$ . Applying localization in T-equivariant Borel-Moore homology, we find that we have a natural isomorphism:

(6.8)

By [Reference WebsterWebb, Theorem 4.3], we have the following result.

Theorem 6.6. There is an equivalence $\mathsf {E} \colon \widehat {\mathscr X}^B(G,N)\rightarrow \widehat {\mathscr A}_{\mathbb Z}^B(G,N) $ compatible with the isomorphisms (6.6) and (6.8), making the following diagram commute

Let $ \lambda , \lambda ^{\prime } $ be the antidominant lifts of $ \gamma , \gamma ^{\prime } \in \tilde {\mathfrak t}_{\mathbb Z} / W $ . The $ P_\lambda / B $ fibre bundle $X_\lambda \to Y_{\gamma }$ implies that the convolution algebra $H^{G}( X_\lambda \times _{Y_{\gamma }}X_\lambda )$ is a copy of the nilHecke algebra for $W^\lambda $ . By [Reference Chriss and GinzburgCG97, Theorem 8.6.7], this algebra acts on the pushforward of the constant sheaf from $X_\lambda $ . Since the nilHecke algebra is a matrix algebra on the commutative algebra $H^*_G(pt)$ , this shows that this pushforward is a sum of $\# W$ copies of the constant sheaf on $ Y_{\gamma }$ . Thus, any primitive idempotent in this nilHecke algebra (in particular, the symmetrizing idempotent $e^{\prime }({\lambda })\in {\mathbb C} W^{\lambda }$ ) gives this constant sheaf. This shows that:

(6.9) $$ \begin{align} \widehat{H}^{{L}}( {}_{\gamma}{\mathbb{Y}}_{{\gamma^{\prime}}}) \cong e^{\prime}({\lambda}) \widehat{H}^{{L}}({}_{\lambda}{\mathbb{X}}_{{\lambda^{\prime}}} )e^{\prime}({\lambda^{\prime}}). \end{align} $$

By [Reference WebsterWeba, Lemma 2.8] (also discussed in the proof of Lemma 5.4), we have a similar formula

(6.10)

Thus, comparing equations (6.9) and (6.10), we have that:

Corollary 6.7. There is an equivalence $ \mathsf {E}\colon \widehat {\mathscr X}(G,N) \rightarrow \widehat {\mathscr A}_{\mathbb Z}(G,N) $ .

Remark 6.8. In [Reference WebsterWeba, proof of Theorem 4.4], following a suggestion of Nakajima, the third author gave a sketch proof of Theorem 6.6 and Corollary 6.7 using Abelianization in equivariant homology.

Remark 6.9. The effect of requiring Z-semisimplicity is encapsulated very cleanly in this theorem. If we consider the weight functors on the category of all modules (rather than restricting to $ Z$ -semisimple ones), we obtain a different category, temporarily denoted $\widehat {\mathscr A}_{\mathbb Z}(G,N)^f$ , whose morphism spaces are given by

A simple Gelfand-Tsetlin module over $\mathcal A$ will factor through one of the quotients $\mathcal A_{\varphi }$ , but there can be extensions of such modules where the centre Z acts nonsemisimply, and thus don’t factor through any such quotient.

In order to fix Corollary 6.7 to work in this setting, we define $ \widehat {\mathscr X}(G,N)^f $ by working with $\tilde G$ -equivariant cohomology instead of G-equivariant so that

$$ \begin{align*}\operatorname{Hom}_{\widehat{\mathscr X}(G,N)^f}(\gamma, \gamma^{\prime}) = \widehat{H}^{\tilde G}( {}_{\gamma}{\mathbb{Y}}_{{\gamma^{\prime}}}).\end{align*} $$

With these definitions, essentially the same argument gives us an equivalence of categories

$$ \begin{align*}\widehat{\mathscr X}(G,N)^f \rightarrow \widehat{\mathscr A}_{\mathbb Z}(G,N)^f. \end{align*} $$

The additional equivariant parameters for the larger group $\tilde G$ capture the action of the nilpotent part of Z, which as we mentioned above might be nontrivial.