2.1.1

We fix once and for all a pinning $(T,B,\psi )$ of $G$. That is, we fix $T \subset B \subset G$ where $T$ is a Cartan and $B$ a Borel subgroup of $G$. In addition, for the unipotent radical $N$ of $B$, we fix a nondegenerate character $\psi :N \to \mathbb {G}_a$.

2.1.2

Given the above data, there is a canonically defined Langlands dual group $\check {G}$, which also comes with a pinning. In particular, we have Cartan and Borel subgroups $\check {T} \subset \check {B} \subset \check {G}$. Again, $\check {N}$ denotes the unipotent radical of $\check {B}$.

2.1.3

The data $T \subset B \subset G$ determine a Borel $B^-$ opposite to $B$, so $B^- \cap B = T$. We denote its radical by $N^-$. The same applies for $\check {G}$.

2.1.4

We denote the appropriate Lie algebras by $\mathfrak {g}, \mathfrak {b}, \mathfrak {n}, \mathfrak {t}, \mathfrak {b}^-, \mathfrak {n}^-, \check {\mathfrak {g}}, \check {\mathfrak {b}}, \check {\mathfrak {n}}, \check {\mathfrak {t}}, \check {\mathfrak {b}}^-$, and $\check {\mathfrak {n}}^-$.

2.1.5

We write $\check {\Lambda }_G$ for the lattice of coweights of $G$, i.e. the cocharacter lattice of $T$, and $\Lambda _G$ for the lattice of weights of $G$, i.e. the character lattice of $T$. We denote the root lattice, i.e. the integral span of the roots, by

\[ Q \subset \Lambda_G. \]

In other words, $Q = \Lambda _{G^\textrm {ad}}$ where $G^\textrm {ad}$ is the adjoint group of $G$. Similarly, we let $\check {Q} \subset \check {\Lambda }_{\check {G}}$ denote the coroot lattice, and one has $\check {Q} = \Lambda _{\check {G}^\textrm {ad}}$.

2.1.6

We let ${\mathcal I}$ denote the set of nodes of the Dynkin diagram of $G$. For $i \in {\mathcal I}$, the corresponding simple roots and coroots are denoted by

\[ \alpha_i \in \Lambda_G \quad \text{and} \quad \check{\alpha}_i \in \check{\Lambda}_G. \]

2.1.7

Our choice of pinning defines a standard involutive anti-homomorphismFootnote ⁵

\[ \tau:\mathfrak{g} \xrightarrow{\sim} \mathfrak{g}. \]

More precisely, for $i \in {\mathcal I}$ let $e_i \in \mathfrak {n}$ be the unique vector of weight $\alpha _i$ such that $\psi (e_i) = 1$. Then $\tau$ is the unique involution such that $\tau ([x,y]) = -[\tau (x),\tau (y)]$, $\tau |_{\mathfrak {t}} = \text {id}_{\mathfrak {t}}$, and

\[ [e_i,\tau(e_i)] = \check{\alpha_i} \quad \text{for } i \in {\mathcal I}. \]

We observe that $\tau$ lifts to an involutive anti-homomorphism on $G$, which we also denote by $\tau$.

2.2.1

For any affine variety $Z$ of finite type, we let $\mathfrak {L} Z$ denote its algebraic loop space and $\mathfrak {L}^+ Z$ its algebraic arc space. The former is an ind-scheme, while the latter is an affine scheme. There is a canonical evaluation map $\mathfrak {L}^+ Z \to Z$ given by evaluation of a jet at the origin.

For an affine algebraic group $Z = H$, $\mathfrak {L} H$ is a group ind-scheme, with $\mathfrak {L}^+H \subset \mathfrak {L} H$ a group subscheme.

2.2.2

The Borel subgroup $B \subset G$ defines the Iwahori subgroup

\[ I := \mathfrak{L}^+ G \times_G B \subset \mathfrak{L}^+ G \subset \mathfrak{L} G. \]

Dually, we have a preferred Iwahori subgroup $\check {I} \subset \mathfrak {L} \check {G}$. We denote the pro-unipotent radical of $I$ by $\mathring {I}$ and note the canonical isomorphism

\[ T \simeq I / \mathring{I}. \]

2.3.1

We let $W_f$ denote the Weyl group of $G$ and recall that $W_f$ is also the Weyl group of $\check {G}$.

2.3.2

The extended affine Weyl group of $G$ is the semidirect product

\[ \tilde{W} := W_f \ltimes \check{\Lambda}_G. \]

The subgroup of $\tilde {W}$ given by

\[ W := W_f \ltimes \check{Q} \]

is the affine Weyl group, where we recall that $\check {Q}$ is the coroot lattice.

2.3.3

Let $S_f \subset W_f$ denote the set of simple reflections $s_i$ for $i \in {\mathcal I}$. Let $S \subset W$ denote the union of $S_f$ with the set of simple affine reflections as in [Reference KacKac90, § 7]. We remind the reader that the simple affine reflections are indexed by simple factors of $G$.

The pairs $(W_f,S_f)$ and $(W,S)$ are Coxeter systems, i.e. Coxeter groups with preferred choices of simple reflections. We recall that the Bruhat order and length function on $W$ each extend in a standard way to $\tilde {W}$.

2.4.1

We let $\operatorname {DGCat}_\textrm {cont}$ denote the symmetric monoidal $\infty$-category of cocomplete DG categories and continuous DG functors as defined in [Reference Gaitsgory and RozenblyumGR17, § I.1.10]. We denote the binary product underlying the symmetric monoidal structure by $-\otimes -$; this is the Lurie tensor product.

For simplicity, we sometimes refer to $\infty$-categories as categories, and similarly for DG categories.

2.4.2

Given a $t$-structure on a DG category ${\mathcal C}$, we let ${\mathcal C}^{\leqslant 0}$ and ${\mathcal C}^{\geqslant 0}$ denote the subcategories of connective and coconnective objects. That is, we use cohomological indexing notation.

We denote the heart of such a $t$-structure by

\[ {\mathcal C}^\heartsuit := {\mathcal C}^{\leqslant 0} \cap {\mathcal C}^{\geqslant 0}. \]

2.5.1

By functoriality, $D(\mathfrak {L} G)$ carries a canonical convolution monoidal structure. We denote the corresponding ($\infty$-)category of DG categories equipped with an action of $D(\mathfrak {L} G)$ by

\[ D(\mathfrak{L} G){-}\mathsf{mod} := D(\mathfrak{L} G){-}\mathsf{mod}(\operatorname{DGCat}_\textrm{cont}). \]

We use similar notation for other group (ind-)schemes such as $\check {I}$ and $\mathfrak {L} N$.

2.6.1

Given a group ind-scheme $H$ and a $D(H)$-module ${\mathcal C}$, we denote its categories of invariants and coinvariants by

\[ {\mathcal C}^H := \mathsf{Hom}_{D(H){-}\mathsf{mod}}(\mathsf{Vect},{\mathcal C}) \quad \text{and} \quad {\mathcal C}_H := \mathsf{Vect} \underset{D(H)}{\otimes} {\mathcal C}, \]

respectively. See [Reference BeraldoBer17] for further discussion.

2.6.2

Similarly, for a multiplicative D-module $\chi$ on $H$, we denote the corresponding categories of twisted invariants and twisted coinvariants by

\[ {\mathcal C}^{H, \chi} \quad \text{and} \quad {\mathcal C}_{H, \chi}. \]

Our multiplicative D-modules will be obtained by one of the following two procedures.

2.6.3

First, any character

\[ \lambda \in \operatorname{Hom}(H,\mathbb{G}_m) \otimes_{\mathbb{Z}} \mathbb{C} \]

determines a character D-module ‘$t^\lambda$’ of $H$. In this case, we denote twisted invariants by ${\mathcal C}^{H,\lambda }$. We apply this construction particularly for $H = T$ and $H = I$.

2.6.4

Similarly, given an additive character

\[ \psi: H \rightarrow \mathbb{G}_a, \]

we obtain a character D-module ‘$e^\psi$’ on $H$. Here we denote twisted invariants by ${\mathcal C}^{H,\psi }$. We apply this construction for $H = \mathfrak {L} N$.

2.6.5

Suppose $H$ is an affine group scheme with pro-unipotent radical $H^u$. We suppose that $H/H^u$ is of finite type.

By [Reference BeraldoBer17], the canonical forgetful map ${\mathcal C}^H \rightarrow {\mathcal C}$ admits a continuous right adjoint $\operatorname {Av}_*^H$. Moreover, there is a canonical equivalence ${\mathcal C}_H \simeq {\mathcal C}^H$ fitting into the following commutative diagram.

(2.1)

The same is true in the presence of a multiplicative D-module $\chi$.

2.6.6

Now suppose ${\mathcal C} \in D(\mathfrak {L} G){-}\mathsf{mod}$. Let $\psi :\mathfrak {L} N \to \mathbb {G}_a$ denote the Whittaker character of $\mathfrak {L} N$.

In this case, [Reference RaskinRas21b, Theorem 2.1.1] provides a canonical equivalence

(2.2)\begin{equation} {\mathcal C}_{\mathfrak{L} N, \psi} \simeq {\mathcal C}^{\mathfrak{L} N, \psi}. \end{equation}

We highlight that this case is more subtle than that of an affine group scheme considered above.

2.6.7

As a final piece of notation, for an ind-scheme $X$ with an action of a group ind-scheme $H$, we use the notation

(2.3)\begin{equation} D(X/H, \chi) := D(X)_{H, \chi}. \end{equation}

By [Reference RaskinRas15b, Proposition 6.7.1], this notation is unambiguous.

We remark that in the setting of either (2.1) or (2.2), these coinvariants coincide with invariants.

2.7.1

We let $\kappa _{\mathfrak {g},c}$ denote the critical level for $G$, i.e. $-\frac {1}{2}$ times the Killing form of $G$. Where $G$ is unambiguous, we simply write $\kappa _c$.

2.7.2

Suppose $G$ is simple. A level $\kappa$ is rational if $\kappa$ is a rational multiple of the Killing form and irrational otherwise. We say that $\kappa$ is positive if $\kappa -\kappa _c$ is a positive rational multiple of the Killing form. We say that a level $\kappa$ is negative if $\kappa$ is not positive or critical. In particular, any irrational level is negative.

For a general reductive $G$, we say that a level $\kappa$ is rational, irrational, positive, or negative if its restrictions to each simple factor are so.

2.7.3

A level $\kappa$ is nondegenerate if $\kappa -\kappa _c$ is nondegenerate as a bilinear form. For such $\kappa$, the dual level $\check {\kappa }$ for $\check {G}$ is the unique nondegenerate level such that the restriction of $\check {\kappa }-\check {\kappa }_{\check {\mathfrak {g}},c}$ to $\mathfrak {t}^*$ and the restriction of $\kappa -\kappa _{\mathfrak {g},c}$ to ${\mathfrak {t}}$ are dual symmetric bilinear forms.

2.7.4

For a simple Lie algebra $\mathfrak {g}$, the basic level $\kappa _{\mathfrak {g},b} = \kappa _b$ is the unique positive level such that the short coroots have squared length 2, i.e.

\[ \min_{i \in {\mathcal I}} \:\kappa_b(\check{\alpha}_i,\check{\alpha}_i) = 2. \]

2.8.1

We recall that the multiplicative twisting defined by $\kappa$ is canonically trivialized on $\mathfrak {L}^+ G$ and $\mathfrak {L} N$. In particular, for

(2.4)\begin{equation} {\mathcal C} \in D_\kappa(\mathfrak{L} G){-}\mathsf{mod} \end{equation}

we can make sense of invariants and coinvariants of $\mathfrak {L}^+ G$ and $\mathfrak {L} N$ with coefficients in ${\mathcal C}$. The same is true for any subgroup, in particular for $I \subset \mathfrak {L}^+ G$. Also, the same applies with a twisting, such as in the Whittaker setup for $\mathfrak {L} N$. We remark that the identification (2.2) of Whittaker invariants and coinvariants [Reference RaskinRas21b] was proved more generally for $D_{\kappa }(\mathfrak {L} G){-}\mathsf{mod}$.

Example 2.8.2 Because $D_\kappa (\mathfrak {L} G)$ carries commuting actions of $D(\mathfrak {L} N)$ and $D(I)$, following the convention of § 2.6.7 we use the notation

\[ D_\kappa(\mathfrak{L} N, \psi \backslash \mathfrak{L} G / I) \]

for the appropriate invariants$\:=\:$coinvariants category.

2.9.1

Let $\mathfrak {L} \mathfrak {g}$ denote the Lie algebra of $\mathfrak {L} G$, considered with its natural inverse limit topology, that is,

\[ \mathfrak{L} \mathfrak{g} = \mathfrak{g}\otimes \mathbb{C}(\!(t)\!) = \operatorname{lim}_n \mathfrak{g} \otimes \mathbb{C}(\!(t)\!)/t^n \mathbb{C}[\![t]\!]. \]

Given a level $\kappa$, one obtains a continuous 2-cocycle $\mathfrak {L} \mathfrak {g} \otimes \mathfrak {L} \mathfrak {g} \to \mathbb {C}$ given by

\[ \xi_1 \otimes \xi_2 \mapsto \operatorname{Res} \kappa(d\xi_1,\xi_2). \]

Here $d$ is the exterior derivative and $\operatorname {Res}$ is the residue. We denote the corresponding central extension by

\[ 0 \rightarrow \mathbb{C} \mathbf{1} \rightarrow \widehat{\mathfrak{g}}_\kappa \rightarrow \mathfrak{L} \mathfrak{g} \rightarrow 0. \]

2.9.2

We denote by $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}^{\heartsuit }$ the abelian category of smooth representations of $\widehat {\mathfrak {g}}_\kappa$ on which the central element $\mathbf {1}$ acts via the identity.

We let $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}$ denote the DG category introduced by Frenkel and Gaitsgory in [Reference Frenkel and GaitsgoryFG09a, §§ 22 and 23]. This DG category is compactly generated and carries a canonical $t$-structure with heart $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}^{\heartsuit }$. However, there are some nonzero objects in

\[ \widehat{\mathfrak{g}}_\kappa{-}\mathsf{mod}^{-\infty} := \bigcap_n \hspace{1mm} \widehat{\mathfrak{g}}_\kappa{-}\mathsf{mod}^{\leqslant -n}, \]

so $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}$ is not the derived category of $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}^{\heartsuit }$.

2.9.3

In [Reference RaskinRas20, §§ 10 and 11], a $D_{\kappa }(\mathfrak {L} G)$-module structure on $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}$ was constructed, enhancing previous constructions of Beilinson and Drinfeld [Reference Beilinson and DrinfeldBD99, § 7] and of Frenkel and Gaitsgory [Reference Frenkel and GaitsgoryFG06b, § 22].

2.9.4

Let $H$ be a sub-group scheme of $\mathfrak {L}^+ G$ of finite codimension, and consider the corresponding category of equivariant objects

\[ \widehat{\mathfrak{g}}_\kappa{-}\mathsf{mod}^{H}. \]

By [Reference RaskinRas21b, Lemma A.35.1], the bounded-below category

\[ \widehat{\mathfrak{g}}_\kappa{-}\mathsf{mod}^{H,+} \]

canonically identifies with the bounded-below derived category of its heart $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}^{H,\heartsuit }$, which consists of Harish-Chandra modules for the pair $(\widehat {\mathfrak {g}}_\kappa, H)$. Moreover, $\widehat {\mathfrak {g}}_\kappa {-}\mathsf{mod}^H$ is compactly generated by inductions of finite-dimensional $H$-modules.

2.9.5

Let $\kappa$ be a level. With an abuse of notation we let $\kappa$ also denote the map $\kappa :\mathfrak {t} \to \mathfrak {t}^*$.

The dot action of $\tilde {W}$ on $\mathfrak {t}^*$ is defined by having $W_f$ act through the usual dot action and having $\check {\Lambda }_G$ act through translations via $-(\kappa - \kappa _c)$; that is, writing $\rho$ for the half-sum of the positive roots, for any $\lambda \in \mathfrak {t}^*$ we have

\begin{align*} w \cdot \lambda & := w(\lambda+\rho)-\rho, \quad w \in W_f \subset \tilde{W}, \\ \check{\mu} \cdot \lambda & := \lambda - (\kappa-\kappa_c)(\check{\mu}), \quad \check{\mu} \in \check{\Lambda} \subset \tilde{W}. \end{align*}

2.9.6

To discuss integral Weyl groups, we need some more standard facts about this dot action. Recall that the affine real coroots of $\widehat {\mathfrak {g}}_\kappa$ form a subset of $\mathfrak {t} \oplus \mathbb {C} \mathbf {1}$. In particular, to such a coroot $\check {\alpha }$ we may associate its classical part $\check {\alpha }_\textrm {cl}$, i.e. its projection to $\mathfrak {t}$. This is a coroot of $\mathfrak {g}$, and in particular we may associate a classical root $\alpha _\textrm {cl} \in \mathfrak {t}^*$.

2.9.7

Via a standard construction, $\check {\alpha }$ acts as an affine linear functional on $\mathfrak {t}^*$, which we denote by $\langle \check {\alpha }, -\rangle$, in such a way that, writing $s_{\check {\alpha }}$ for the associated reflection in $\tilde {W}$, one has

\[ s_{\check{\alpha}} \cdot \lambda = \lambda - \langle\check{\alpha}, \lambda + \rho \rangle \alpha_\textrm{cl}, \quad \lambda \in \mathfrak{t}^*. \]

Briefly, this arises via restricting a linear action of $\tilde {W}$ on $(\mathfrak {t} \oplus \mathbb {C} \mathbf {1})^*$ to the affine hyperplane of functionals whose pairing with $\mathbf {1}$ is 1. We refer the reader to [Reference DhillonDhi21, §§ 3.1 and 3.4], where this is reviewed in greater detail.

3.1.1

Suppose $\lambda \in \mathfrak {t}^{*}$ is a weight of $\mathfrak {g}$. Let us denote the block of Category ${\mathcal O}$ for $\widehat {\mathfrak {g}}_\kappa$ containing the Verma module $M_\lambda$ by

\[ {\mathcal O}_{\kappa,\lambda} \subset \widehat{\mathfrak{g}}_{\kappa}{-}\mathsf{mod}^{\mathring{I}}. \]

In § 3.2.3, we recall that $\lambda$ determines a subgroup $W_{\lambda } \subset W$, its integral Weyl group.Footnote ⁶ This is a Coxeter group, i.e. it comes equipped with a set of simple reflections.

We make important use of Theorem 3.5.2, which is due to Fiebig [Reference FiebigFie06], following earlier work of Soergel [Reference SoergelSoe90]. This result asserts that if $\lambda$ is antidominant,Footnote ⁷ ${\mathcal O}_{\kappa,\lambda }$ is determined as a category by the data of (i) the Coxeter group $W_{\lambda }$ and (ii) the subgroup

(3.1)\begin{equation} W_\lambda^\circ\subset W_\lambda \end{equation}

stabilizing $\lambda$ under the dot action of $W$ on $\mathfrak {t}^*$; cf. § 2.9.5.

In Theorem 3.2.7, we find a block decomposition for the Whittaker category $\mathsf {Whit}_{\kappa }^{\operatorname {aff}}$ on the affine flag variety for $G$. These blocks are indexed by $W_{\mathfrak {g},\kappa }$-orbits in $\check {\Lambda }_G = \Lambda _{\check {G}}$.

In Theorem 3.3.3, we show that up to varying $\kappa$ by an integral translate (which does not affect the Whittaker category), its neutral block is equivalent to an integral block of Category ${\mathcal O}$ for $\widehat {\mathfrak {g}}_{-\kappa }$. We generalize this to general blocks at goodFootnote ⁸ levels in Corollary 3.4.8. These identifications preserve the natural highest-weight structures on both sides. From now on, we assume that $\kappa$ is good.

3.1.4

For each block of $\mathsf {Whit}_{\kappa }^{\operatorname {aff}}$, we explicitly compute the combinatorial datum (3.1) of the corresponding block of Category ${\mathcal O}$ for the corresponding Kac–Moody algebraFootnote ⁹ and provide a form of Langlands duality for this datum.

Essentially by construction, for any such block the corresponding integral Weyl group is $W_{\mathfrak {g},\kappa }$, with simple reflections as indicated above. In Theorem 3.5.6, we construct an isomorphism $W_{\mathfrak {g},\kappa } \simeq W_{\check {\mathfrak {g}},\check {\kappa }}$ preserving simple reflections, and that is the identity on $W_f$.Footnote ¹⁰

Note that for any block of ${\mathcal O}$ for the Kac–Moody algebra, the corresponding integral Weyl group canonically acts on the set of isomorphism classes of simple objects in this block. Therefore, the above considerations provide an action of $W_{\check {\mathfrak {g}},\check {\kappa }}$ on the set of isomorphism classes of simple objects in $\mathsf {Whit}_{\kappa }^{\operatorname {aff}}$, which is canonically identified with $\check {\Lambda }_G = \Lambda _{\check {G}}$. Again, by construction, this action coincides with the dot action of § 2.9.5, but for $\check {G}$ rather than $G$.

In Corollary 3.2.11 and in the proof of Theorem 3.6.1, we check that simple objects of $\mathsf {Whit}_{\kappa }^{\operatorname {aff}}$ corresponding to antidominant weights of $\check {G}$ are also standard objects for the highest-weight structure on this category. These observations amount to matching the data (3.1) with the data of affine category ${\mathcal O}$ for $\check {G}$, completing the proof of Theorem 1.3 in the Iwahori case.

3.1.5

In § 3.7, under the hypotheses of this section, we deduce the parahoric version of Theorem 1.3 from the Iwahori version. In particular, this includes the spherical version of the theorem.

We do this by identifying the parahoric categories as fullFootnote ¹¹ subcategories of the corresponding Iwahori categories and then identifying the essential images under the isomorphism of Theorem 3.6.1.

3.2.1

To do so, it will be useful to simultaneously consider the case of twisted Whittaker categories. Thus, we fix $\lambda \in \mathfrak {t}^*$ and a level $\kappa$ and consider

\[ \mathsf{Whit}_{\lambda}^{\operatorname{aff}} := D_{\kappa}(\mathfrak{L} N,\psi \backslash \mathfrak{L} G / I,-\lambda). \]

While these DG categories depend on both $\lambda$ and $\kappa$, we will study them with fixed $\kappa$ and varying $\lambda$, and for this reason we suppress $\kappa$ for ease of notation.

3.2.2

For indexing reasons, it will be convenient to rewrite this category as follows. Consider the automorphism of $\mathfrak {L} G$ given by

\[ g \mapsto \tau(g) \check{\rho}(t^{{-}1}), \]

where $\tau$ is as in § 2.1.7. This induces an equivalence

(3.2)\begin{equation} D_\kappa(\mathfrak{L} N, \psi \backslash \mathfrak{L} G / I,-\lambda) \simeq D_{-\kappa}(I,\lambda \backslash \mathfrak{L} G /\mathfrak{L} N^-, \psi) \end{equation}

where the $\psi$ on the right-hand side denotes a nondegenerate character of $\mathfrak {L} N^-$ of conductor 1. In the following discussion, we will think of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ via the latter expression.

3.2.3

It will be important for us to consider $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ as a module for the following version of the affine Hecke category.

Let $W_\lambda$ denote the integral Weyl group of $\lambda$.Footnote ¹² Recall that this is the subgroup of $W$ that generated the reflections $s_{\check {\alpha }}$ corresponding to affine coroots $\check {\alpha }$ satisfying

\[ \langle \check{\alpha}, \lambda \rangle \in \mathbb{Z}, \]

where the pairing is via the action at level $-\kappa$ as in § 2.9.7. For $w \in W_\lambda$, write $j_{w, !*}$ for the intermediate extension of the simple object of

\[ D_{-\kappa}(I, \lambda \backslash IwI / I, -\lambda)^\heartsuit. \]

We define ${\mathcal H}_\lambda$ to be the full subcategory of

(3.3)\begin{equation} D_{-\kappa}(I, \lambda \backslash \mathfrak{L} G / I, -\lambda) \end{equation}

generated under colimits and shifts by the objects

\[ j_{w, !*} \quad \text{for} \ w \in W_\lambda. \]

By [Reference Lusztig and YunLY20, § 4],Footnote ¹³ ${\mathcal H}_\lambda$ is closed under convolution and so admits a unique monoidal DG structure for which its embedding into (3.3) is monoidal. In addition, loc. cit. shows that ${\mathcal H}$ is the neutral block of (3.3), i.e. it is the minimal direct summand containing the identity element.

3.2.4

Recall that the Iwasawa decomposition provides $\mathfrak {L} G$ with a stratification by the double cosets

\[ I w \mathfrak{L} N^- \quad \text{for} \ w \in \tilde{W}. \]

Let $\tilde {W}^f \subset \tilde {W}$ denote the subset of elements of minimal length in their left $W_f$-cosets. We now describe the affine Whittaker category for a single stratum.

Lemma 3.2.5 For $w \notin \tilde {W}^f$, the affine Whittaker category on $Iw \mathfrak {L} N^-$ vanishes, i.e.

\[ D_{-\kappa}(I, \lambda \backslash I w \mathfrak{L} N^-{/}\mathfrak{L} N^-, \psi) \simeq 0. \]

For $w \in \tilde {W}^f$, $!$-restriction to any closed point gives an equivalence

\[ D_{-\kappa}(I, \lambda \backslash I w \mathfrak{L} N^-{/} \mathfrak{L} N^-, \psi) \simeq \mathsf{Vect}. \]

For $w \in \tilde {W}^f$ we denote the corresponding standard, simple, and costandard objects of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ by

\[ j_{w, !}^\psi, \quad j_{w, !*}^\psi, \quad \text{and} \quad j_{w, *}^\psi. \]

Explicitly, under the identification with $\mathsf {Vect}$ above, they correspond to the relevant extensions of $\mathbb {C}[-\ell (ww_{\circ })] \in \mathsf {Vect}$, where $\ell$ denotes the length function on $\tilde {W}$ and $w_\circ$ the longest element of $W_f$.

3.2.6

We next obtain the block decomposition of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$. To state it, for any double coset

\[ W_\lambda y W_f \]

we write $\mathsf {Whit}_{\lambda,y}^{\operatorname {aff}}$ for the full subcategory of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ generated under colimits and shifts by the objects

\[ j_{x, *}^\psi \quad \text{for } x \in W_\lambda y W_f \cap {\tilde{W}^f}. \]

Theorem 3.2.7 Each $\mathsf {Whit}_{\lambda, y}^{\operatorname {aff}}$ is preserved by the action of ${\mathcal H}_\lambda$, and the direct sum of inclusions yields an ${\mathcal H}_\lambda$-equivariant equivalence

(3.4)\begin{equation} \underset{ y \in W_\lambda \backslash \tilde{W} / W_f } {\bigoplus} \mathsf{Whit}_{\lambda, y}^{\operatorname{aff}} \to \mathsf{Whit}_\lambda^{\operatorname{aff}}. \end{equation}

Proof. For $w \in \tilde {W}$ consider the corresponding standard, simple, and costandard objects

\[ j_{w, !}, \quad j_{w, !*}, \quad \text{and} \quad j_{w, *} \quad \text{of} \quad D_{-\kappa}(I, w \cdot \lambda \backslash \mathfrak{L} G / I, -\lambda)^{\heartsuit}. \]

We use $-\overset {I}{\star }-$ to denote the convolution functor

\[ D_{-\kappa}(\mathfrak{L} G / I, -\lambda) \otimes D_{-\kappa}(I, \lambda \backslash \mathfrak{L} G) \rightarrow D_{-\kappa}(\mathfrak{L} G) \]

and the induced functor

(3.5)\begin{equation} D_{-\kappa}(I, w \cdot \lambda \backslash \mathfrak{L} G / I, -\lambda ) \otimes D_{-\kappa}(I, \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi) \rightarrow D_{-\kappa}(I, w \cdot \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi). \end{equation}

For any $w \in \tilde {W}$, if with an abuse of notation we denote its image in $\tilde {W}^f \xrightarrow {\sim } \tilde {W}/W_f$ again by $w$, we claim that there exist equivalences

(3.6)\begin{equation} j_{w, !} \overset{I}{\star} j_{e, *}^\psi \simeq j_{w, !}^\psi \quad \text{and} \quad j_{w, *} \overset{I}{\star} j_{e, *}^\psi \simeq j_{w, *}^\psi. \end{equation}

Note that in (3.6), the object $j_{e, *}^\psi$ belongs to $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$, whereas $j_{w, !}^\psi$ and $j_{w, *}^\psi$ belong to $\mathsf {Whit}_{w \cdot \lambda }^{\operatorname {aff}}$.

We split the proof of (3.6) into several cases. First, suppose $w$ lies in $\tilde {W}^f$. Then the second identity in (3.6) follows from the observation that the convolution map

(3.7)\begin{equation} IwI \overset{I}{\times} I\mathfrak{L} N^- \rightarrow Iw \mathfrak{L} N^- \end{equation}

is an isomorphism. The first identity for such $w$ then follows by the cleanness of $j_{e, *}^\psi$ and the ind-properness of the multiplication map

\[ \mathfrak{L} G \overset{I}{\times} \mathfrak{L} G \rightarrow \mathfrak{L} G. \]

Next, suppose $w = s$ is a simple reflection of $W_f$. The image of the convolution map as in (3.7) is contained in the locally closed sub-ind-scheme of $\mathfrak {L} G$ corresponding to the strata $I \mathfrak {L} N^- \cup I s \mathfrak {L} N^-$. As the only stratum of its closure that supports Whittaker sheaves is $I \mathfrak {L} N^-$, it is enough to compute the $!$-restrictions of our convolutions to the identity

\[ i: e \rightarrow \mathfrak{L} G. \]

Let us write $\iota$ for the involution $g \mapsto g^{-1}$ on $\mathfrak {L} G$. By base change we may compute the $!$-fibre as

\[ i^!(j_{s,*} \overset{I}{\star} j_{e,*}^{\psi}) \simeq \Gamma_{dR}(I \backslash \mathfrak{L} G, \iota_* j_{s, *} \overset{!}{\otimes} j_{e, *}^\psi) \simeq \Gamma_{dR}(\mathbb{P}^1\setminus \{0,\infty\},\text{`}e^t\text{'}[-\ell(w_0)]) \simeq \mathbb{C}[- \ell({w_\circ})], \]

as desired. A similar calculation on $\mathbb {P}^1$ yields the first identity in (3.6) for $w = s$.

Finally, for general $w \in \tilde {W}$, we write $w = w^f w_f$ for $w^f \in \tilde {W}^f$ and $w_f \in W_f$. Choosing a reduced expression for $w_f$ and the corresponding factorizations of $j_{w, !}$ and $j_{w, *}$, we reduce to the cases considered above for twists in $\tilde {W} \cdot \lambda$. Given (3.6), the assertions of the theorem follow by the same argument as for [Reference Lusztig and YunLY20, Proposition 4.11].

3.2.8

Having obtained the block decomposition of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$, we now record some properties of each block as an ${\mathcal H}_\lambda$ module. We begin with some relevant combinatorics.

Recall that $W_\lambda$ is a Coxeter group. Write $\check {\Phi }^+$ for the positive real coroots and consider the subset

\[ \check{\Phi}^+_\lambda = \{ \check{\alpha} \in \check{\Phi}^+: \langle \check{\alpha}, \lambda \rangle \in \mathbb{Z} \}. \]

With this, a reflection $s_{\check {\alpha }}$ of $W_\lambda$ is simple if and only if $\check {\alpha }$ is not expressible as a sum of two other elements of $\check {\Phi }^+_\lambda$.

Lemma 3.2.9 Each double coset $W_\lambda y {W}_{f}$ contains a unique element that is minimal with respect to the Bruhat order. In particular, each $W_\lambda$ orbit on $\tilde {W} / W_{f}$ contains a unique element $yW_f$ that is minimal with respect to the Bruhat order. Moreover, its stabilizer

(3.8)\begin{equation} y W_f y^{{-}1} \cap W_\lambda \end{equation}

is a parabolic subgroup of $W_\lambda$.

Proof. Let $y$ be any element of minimal length in $W_\lambda y W_{f}$. We first claim that $y \leqslant w_\lambda y$ for every $w_\lambda \in W_\lambda$. Indeed, for any reflection $s_{\check {\alpha }}$ of $W_\lambda$ with corresponding positive coroot $\check {\alpha }$, it follows from the minimal length of $y$ that

(3.9)\begin{equation} s_{\check{\alpha}} y > y, \end{equation}

i.e. $y^{-1}(\check {\alpha }) > 0$. The claimed minimality now follows by induction on the length of an element of $W_\lambda$ with respect to its Coxeter generators.

We next show that $y \leqslant w_\lambda y w_f$ for any $w_f$ in $W_f$. However, it is clear that $y \leqslant ys$ for any simple reflection $s$ of $W_f$. This implies that for any $w \in \tilde {W}$, we have $y \leqslant w$ if and only if $y \leqslant ws_j$. Applying this and a straightforward induction on the length to $w = w_\lambda y$ yields the claim.

It remains to show that (3.8) is a parabolic subgroup of $W_\lambda$. To see this, consider $W_{y^{-1} \lambda } = y^{-1} W_\lambda y$. The positivity property (3.9) of $y$ implies that conjugation by $y^{-1}$ defines an isomorphism of Coxeter systems

\[ W_\lambda \simeq W_{y^{{-}1} \lambda}, \]

i.e. exchange of their simple reflections. It is therefore enough to show that $W_f \cap W_{y^{-1} \lambda }$ is a parabolic subgroup of $W_{y^{-1} \lambda }$, which is straightforward; cf. the proof of Lemma 8.8 in [Reference DhillonDhi21].

3.2.10

Combining Lemma 3.2.9 and Theorem 3.2.7, we obtain the following result.

Corollary 3.2.11 For a double coset $W_\lambda y W_f$ with minimal element $y$, the corresponding object of $\mathsf {Whit}_{\lambda, y}^{\operatorname {aff}}$ is clean, i.e.

\[ j_{y, !}^\psi \simeq j_{y, !*}^\psi \simeq j_{y, *}^\psi. \]

In the following proposition, we collect some properties of the action of ${\mathcal H}_\lambda$ on $j_{y, !}^\psi$. To state them, let us denote the set of elements of $W_\lambda$ of minimal length in their left cosets with respect to the parabolic subgroup (3.8) by

(3.10)\begin{equation} W_\lambda^f. \end{equation}

We additionally recall that $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ carries a canonical $t$-structure; namely, an object ${\mathcal M}$ is coconnective, i.e. lies in $\mathsf {Whit}_{\lambda }^{\operatorname {aff}, \geqslant 0}$, if and only if

\[ \operatorname{Hom}( j_{w, !}^\psi, {\mathcal M}) \in \mathsf{Vect}^{\geqslant 0} \quad \text{for } w \in \tilde{W}^f. \]

Such a $t$-structure may be seen directly via gluing of $t$-structures stratum by stratum, and coincides with the general construction in § 5.2 and Appendix B of [Reference RaskinRas21b]; cf. loc. cit. Remark B.7.1.

Proposition 3.2.12 For $y$ as in Corollary 3.2.11 and for any $w$ in $W_\lambda$, there are equivalences

(3.11)\begin{equation} j_{w, !} \overset{I}{\star} j_{y, *}^\psi \simeq j_{wy, !}^\psi \quad \text{and} \quad j_{w, *} \overset{I}{\star} j_{y, *}^\psi \simeq j_{wy, *}^\psi. \end{equation}

Moreover, if $w$ lies in $W_\lambda ^f$, then convolution with $j_{y, *}^\psi$ yields an isomorphism of lines

(3.12)\begin{equation} \operatorname{Hom}_{D_{-\kappa}(I, \lambda \backslash \mathfrak{L} G / I, -\lambda)^\heartsuit}(j_{w, !}, j_{w, *}) \simeq \operatorname{Hom}_{\mathsf{Whit}^{\operatorname{aff}}_{\lambda}}(j_{wy, !}^\psi, j_{wy, *}^\psi). \end{equation}

Proof. Both assertions follow from (3.6) by standard arguments. Specifically, the proof of Proposition 5.2 in [Reference Lusztig and YunLY20] yields (3.11). It follows from (3.11) that convolution with $j_{y, *}^\psi$ is $t$-exact.

Let us show that the latter observation implies (3.12). For any $u \in W_\lambda$, recall that $j_{u, !*}$ denotes the simple quotient of $j_{u, !}$. By $t$-exactness, we obtain a surjection

(3.13)\begin{equation} j_{uy, !}^\psi \overset{(3.11)}{\simeq} j_{u, !} \overset{I} \star j_{y, *}^\psi \rightarrow j_{u, !*} \overset{I}{\star} j_{y, *}^\psi \rightarrow 0. \end{equation}

For an element $w$ of $W_\lambda ^f$, consider the tautological exact sequence

\[ 0 \rightarrow K \rightarrow j_{w, !} \rightarrow j_{w, !*} \rightarrow 0. \]

By definition, $K$ admits a filtration by intermediate extensions $j_{u, !*}$ for $u < w$ in the Bruhat order on $W_\lambda$. Convolving with $j_{y, *}^\psi$, by $t$-exactness we again obtain an exact sequence

\[ 0 \rightarrow K \overset I \star j_{y, *}^\psi \rightarrow j_{wy, !}^\psi \rightarrow j_{w, !*} \overset I \star j_{y, *}^\psi \rightarrow 0. \]

By our assumption on $w$, it follows from (3.13) that $[K \overset I \star j_{y, *}^\psi : j_{wy, !*}^\psi ] = 0$. In particular, $j_{w, !*} \overset I \star j_{y, *}^\psi$ is nonzero.

This nonvanishing and the $t$-exactness of convolution with $j_{y, *}^\psi$ imply that the intermediate extension sequence $j_{w, !} \twoheadrightarrow j_{w, !*} \hookrightarrow j_{w, *}$ yields a sequence of nonzero maps

\[ j_{w, !} \overset I \star j_{y, *}^\psi \twoheadrightarrow j_{w, !*} \overset I \star j_{y, *}^\psi \hookrightarrow j_{w, *} \overset I \star j_{y, *}^\psi. \]

This verifies (3.12) (and also shows that $j_{w, !*} \overset I \star j_{y, *}^\psi \simeq j_{wy, !*}^\psi$).

Remark 3.2.13 One can show that the clean objects $j_{y, !*}^\psi$ constructed above are the only clean extensions in $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$. Since we do not use this fact, we only sketch the proof. Specifically, write $\leqslant$ for the Bruhat order on $W_\lambda ^f$. One can in fact show that for any $w, x \in W_\lambda ^f$, the space of intertwining operators

\[ \operatorname{Hom}_{\mathsf{Whit}_{\lambda}^{\operatorname{aff}, \heartsuit}}( j_{wy, !}^\psi, j_{xy, !}^\psi) \]

vanishes unless $w \leqslant x$, in which case it is one-dimensional and consists of embeddings. This may be deduced from (3.11) and the analogous classification of intertwining operators between standard objects of ${\mathcal H}_\lambda$. The latter may be proved directly or deduced via localization from the Kac–Kazhdan theorem on homomorphisms between Verma modules.

3.3.1

Consider the category of $(I, \lambda )$-integrable modules for $\widehat {\mathfrak {g}}_{-\kappa }$, which we denote by

(3.14)\begin{equation} \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{I, \lambda}. \end{equation}

Recall that this is compactly generated by the Verma modules $M_\theta \text { for } \theta \in \lambda + \Lambda _G,$ and note that the integral Weyl group of any such $\theta$ is $W_\lambda$.

3.3.2

We identify the desired blocks by matching their combinatorics to that of $\mathsf {Whit}_{e, \lambda }^{\operatorname {aff}}$ as follows. Suppose $M_\Theta$ is an irreducible Verma module in (3.14) such that the stabilizer of $\Theta$ under the dot action of $W_\lambda$ is given by

\[ W_\lambda \cap W_f. \]

Consider the corresponding block of (3.14), i.e. the full subcategory compactly generated by $M_{\theta }$ for $\theta \in W_\lambda \cdot \Theta$, and denote it by

\[ \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{I, \lambda}_\Theta. \]

Theorem 3.3.3 There is a canonical ${\mathcal H}_\lambda$-equivariant and $t$-exact equivalence

(3.15)\begin{equation} \mathsf{Whit}_{e, \lambda}^{\operatorname{aff}} \simeq \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{I, \lambda}_\Theta. \end{equation}

Remark 3.3.4 For emphasis, by definition of $\Theta$, the integral Weyl group of this block is $W_{\lambda }$, and the stabilizer of $\Theta$ in $W_{\lambda }$ is $W_{\lambda } \cap W_f$. In particular, on both sides of (3.15) one finds highest-weight categories whose highest weights are indexed by

\[ W_\lambda/(W_\lambda \cap W_f) \simeq W_\lambda^f; \]

for $\mathsf {Whit}_{e, \lambda }^{\operatorname {aff}}$ compare (3.10) and (3.8) and recall that we are considering the neutral block, i.e. specializing to $y = e$ in (3.8).

Proof of Theorem 3.3.3 To produce an ${\mathcal H}_\lambda$-equivariant functor, we will construct a $D_{-\kappa }(\mathfrak {L} G)$-equivariant functor and then pass to $(I,\lambda )$-equivariant objects. After further projecting onto a block of (3.14), we will show that this is the sought-for equivalence.

We begin by constructing an $D_{-\kappa }(\mathfrak {L} G)$-equivariant functor

\[ \mathsf{F}: D_{-\kappa}(\mathfrak{L} G / \mathfrak{L} N^-, \psi) \rightarrow \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}. \]

To do so, recall that for any $D_{-\kappa }(\mathfrak {L} G)$-module ${\mathcal C}$ one has a canonical equivalence

(3.16)\begin{equation} \mathsf{Hom}_{D_{-\kappa}(\mathfrak{L} G){-}\mathsf{mod}}( D_{-\kappa}(\mathfrak{L} G / \mathfrak{L} N^-, \psi), {\mathcal C}) \simeq {\mathcal C}^{\mathfrak{L} N^-, \psi}. \end{equation}

Explicitly, if we write $\operatorname {ins} \delta$ for the insertion of the delta function at the identity of $\mathfrak {L} G$ into Whittaker coinvariants, the equivalence (3.16) is given by evaluation at $\operatorname {ins} \delta$. Applying this to ${\mathcal C} \simeq \widehat {\mathfrak {g}}_{-\kappa }{-}\mathsf{mod}$, we obtain via the affine Skryabin equivalence (cf. [Reference RaskinRas21b])

(3.17)\begin{equation} \mathsf{Hom}_{D_{-\kappa}(\mathfrak{L} G){-}\mathsf{mod}}( D_{-\kappa}(\mathfrak{L} G / \mathfrak{L} N^-, \psi), \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}) \simeq \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{\mathfrak{L} N^-, \psi} \simeq {\mathcal W}_{-\kappa}{-}\mathsf{mod}. \end{equation}

Therefore, to produce $\mathsf {F}$ we must specify a module for the ${\mathcal W}_{-\kappa }$-algebra.

We do so as follows. Write $\operatorname {Zhu}({\mathcal W}_{-\kappa })$ for the Zhu algebra of ${\mathcal W}_{-\kappa }$ and $Z\mathfrak {g}$ for the center of the universal enveloping algebra of $\mathfrak {g}$. Let us normalize the identification

\[ \operatorname{Zhu}({\mathcal W}_{-\kappa}) \simeq Z \mathfrak{g} \]

as in [Reference DhillonDhi21]. Writing $\chi (\Theta )$ for the character of $Z \mathfrak {g}$ corresponding to $\Theta$, i.e. via its action on the Verma module for $\mathfrak {g}$ with highest weight $\Theta$, consider the associated local cohomology $Z\mathfrak {g}$-module $\textit {Ri}_{\chi (\Theta )}^! Z \mathfrak {g}$. We will take $\mathsf {F}$ to be associated to the corresponding ${\mathcal W}_{-\kappa }$-module

(3.18)\begin{equation} \operatorname{pind}_{\operatorname{Zhu}({\mathcal W}_{-\kappa})}^{{\mathcal W}_{-\kappa}} \textit{Ri}_{\chi(\Theta)}^! Z \mathfrak{g}, \end{equation}

where $\operatorname {pind}$ denotes the standard induction from Zhu algebra modules to vertex algebra modules, i.e. the left adjoint to the functor of taking singular vectors.

Passing to $(I, \lambda )$-equivariant objects, we obtain a $D_{-\kappa }(I, \lambda \backslash \mathfrak {L} G / I, -\lambda )$-equivariant functor

\[ \mathsf{F}: D_{-\kappa}( I, \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi) \rightarrow \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{I, \lambda}. \]

Let us determine $\mathsf {F}(j_{e, *}^\psi )$. To do so, write $\mathbb {C}_\psi$ for the one-dimensional representation of $\mathfrak {n}^-$ associated to the additive character $\psi$ of $\mathfrak {L} N^-$. As we will explain in more detail below, we then have

(3.19)\begin{align} \operatorname{Av}^{I, \lambda}_{*} \operatorname{pind}_{\operatorname{Zhu}({\mathcal W}_{-\kappa})}^{{\mathcal W}_{-\kappa}} \textit{Ri}_{\chi(\Theta)}^! Z \mathfrak{g} &\simeq \operatorname{Av}^{I, \lambda}_{*} \operatorname{pind}_{\mathfrak{g}}^{\widehat{\mathfrak{g}}_{-\kappa}} (\textit{Ri}^!_{\chi(\Theta)} Z\mathfrak{g} \underset{Z \mathfrak{g}}{\otimes} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi) \end{align}

(3.20)\begin{align} & \simeq \operatorname{pind}_{\mathfrak{g}}^{\widehat{\mathfrak{g}}_{-\kappa}} \operatorname{Av}^{B, \lambda}_{*} (\textit{Ri}^!_{\chi(\Theta)} Z \mathfrak{g} \underset{Z \mathfrak{g}}{\otimes} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi) \end{align}

(3.21)\begin{align} & \simeq \bigoplus_{ \xi \in (W_f \cdot \Theta) \cap \lambda + \Lambda_G} M_\xi \otimes \det( \mathfrak{b}^*[{-}1]). \end{align}

To see the first identification, (3.19), one may use that (3.18) is an iterated extension of Verma modules and in particular arises from the first step of the adolescent Whittaker filtration of ${\mathcal W}_{-\kappa }{-}\mathsf{mod}$; cf. [Reference RaskinRas21b].

For the second identification, (3.20), if we write $K_1$ for the first congruence subgroup of $\mathfrak {L}^+ G$, one has a corresponding factorization

\[ \operatorname{Av}_{*}^{I, \lambda} \simeq \operatorname{Av}_{*}^{K_1, \lambda} \circ \operatorname{Av}_{*}^{B, \lambda}. \]

The second identification then follows from the $K_1$-integrability of a parabolically induced module, the pro-unipotence of $K_1$ (cf. Theorem 4.3.2 of [Reference BeraldoBer17]), and the $D(G)$-equivariance of $\operatorname {pind}_{\mathfrak {g}}^{\widehat {\mathfrak {g}}_{-\kappa }}$.

For the third identification, (3.21), for any $\xi \in \lambda + \Lambda _G$ let us write $M_{\xi, \mathfrak {g}}$ for the corresponding Verma module for $\mathfrak {g}$. One has by adjunction

\[ \operatorname{Hom}_{\mathfrak{g}{-}\mathsf{mod}^{B, \lambda}}(M_{\xi, \mathfrak{g}}, \operatorname{Av}^{B, \lambda}_{*} \textit{Ri}^!_{\chi(\Theta)} Z \mathfrak{g} \underset{Z \mathfrak{g}}{\otimes} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi) \simeq \operatorname{Hom}_{\mathfrak{g}{-}\mathsf{mod}}(M_{\xi, \mathfrak{g}}, \textit{Ri}^!_{\chi(\Theta)} Z \mathfrak{g} \underset{Z \mathfrak{g}}{\otimes} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi). \]

The latter vanishes unless $\xi \in W_f \cdot \Theta$, in which case we continue the above to

\[{\simeq} \operatorname{Hom}_{\mathfrak{g}{-}\mathsf{mod}}(M_{\xi, \mathfrak{g}}, \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi) \simeq \operatorname{Hom}_{\mathfrak{b}{-}\mathsf{mod}}(\mathbb{C}_\xi, \operatorname{Res}_{\mathfrak{g}}^{\mathfrak{b}} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi) \simeq \det(\mathfrak{b}^*[{-}1]), \]

where in the last step one uses the canonical identification

\[ U(\mathfrak{b}) \simeq \operatorname{Res}_{\mathfrak{g}}^{\mathfrak{b}} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi. \]

By our assumption on $\Theta$, each $M_{\xi, \mathfrak {g}}$ for $\xi \in (W_f \cdot \Theta ) \cap \lambda + \Lambda _G$ generates a block of $\mathfrak {g}{-}\mathsf{mod}^{B, \lambda }$ equivalent to $\mathsf {Vect}$; thus we have shown that

\[ \operatorname{Av}_{*}^{B, \lambda} \textit{Ri}^!_{\chi(\Theta)} Z \mathfrak{g} \underset{Z \mathfrak{g}}{\otimes} \operatorname{ind}_{\mathfrak{n}^-}^{\mathfrak{g}} \mathbb{C}_\psi \simeq \bigoplus_{ \xi \in (W_f \cdot \Theta) \cap \lambda + \Lambda_G} M_{\xi, \mathfrak{g}} \otimes \det(\mathfrak{b}^*[{-}1]). \]

The identity (3.21) then follows by applying $\operatorname {pind}_{\mathfrak {g}}^{\widehat {\mathfrak {g}}_{-\kappa }}$.

The projection of the sum of Verma modules (3.21) onto $\widehat {\mathfrak {g}}_{-\kappa }{{-}\mathsf{mod}}^{I, \lambda }_\Theta$ picks out the summand

\[ M_\Theta \otimes \det(\mathfrak{b}^*[{-}1]); \]

cf. [Reference DhillonDhi21, Lemma 8.7]. Accordingly, we consider the composition

(3.22)\begin{equation} D_{-\kappa}( I, \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi) \xrightarrow{ \mathsf{F}\,\otimes \,\det( \mathfrak{b}[1]) [\dim N^-]} \widehat{\mathfrak{g}}_{-\kappa}{{-}\mathsf{mod}}^{I, \lambda} \rightarrow \widehat{\mathfrak{g}}_{-\kappa}{{-}\mathsf{mod}}^{I, \lambda}_\Theta, \end{equation}

where the latter map is projection onto the block. Note that while the blocks of (3.14) are in general not preserved by $D_{-\kappa }( I, \lambda \backslash \mathfrak {L} G / I, -\lambda )$, they are preserved by ${\mathcal H}_\lambda$. In particular, by construction the composition (3.22), which we denote by $\Psi$, is an ${\mathcal H}_\lambda$-equivariant functor which sends $j_{e, *}^\psi$ to $M_\Theta$.

It remains to show that $\Psi$ is an equivalence. By (3.11) and ${\mathcal H}_\lambda$-equivariance, we obtain identifications

\[ \Psi( j_{w, !}^\psi) \simeq \Psi(j_{w, !} \overset{I}{\star} j_{e, *}^\psi ) \simeq j_{w, !} \overset{I}{\star} \Psi(j_{e, *}^\psi) \simeq M_{w \cdot \Theta} \quad \text{for} \ w \in W_\lambda, \]

where the last identity is a standard consequence of Kashiwara–Tanisaki localization at a negative level [Reference Kashiwara and TanisakiKT96].Footnote ¹⁴ Similarly, if for $\theta \in \lambda + \Lambda$ we denote the contragredient dual of $M_\theta$ by $A_\theta$, we obtain identifications

\[ \Psi(j_{w, *}^\psi) \simeq \Psi(j_{w, *} \overset{I}{\star} j_{e, *}^\psi) \simeq j_{w, *} \overset{I}{\star} \Psi( j_{e, *}^\psi) \simeq A_{w \cdot \Theta} \quad \text{for} \ w \in W_\lambda. \]

Recall that the (co)standard objects $j_{w, !}^\psi$ and $j_{w, *}^\psi$ of $\mathsf {Whit}_{e, \lambda }$ are indexed by the minimal-length coset representatives of $W_\lambda$ with respect to its parabolic subgroup $W_\lambda \cap W_f$, i.e.

\[ W_\lambda^f, \]

and either collection compactly generates $\mathsf {Whit}_{e, \lambda }$.

Similarly, by our assumption on the stabilizer of $\Theta$ and the irreducibility of $M_\Theta$, the collections of (co)standard objects $M_{w \cdot \Theta }$ and $A_{w \cdot \Theta }$, for $w \in W_\lambda ^f,$ each compactly generates $\widehat {\mathfrak {g}}_{-\kappa }{-}\mathsf{mod}^{I, \lambda }_\Theta$.

It is therefore enough to check that for any $y, w \in W_\lambda ^f$, the map

\[ \Psi: \operatorname{Hom}_{\mathsf{Whit}_{\lambda}^{\operatorname{aff}}}( j_{y, !}^\psi, j_{w, *}^\psi) \rightarrow \operatorname{Hom}_{\widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{I, \lambda}}( M_{y \cdot \Theta}, A_{y \cdot \Theta}) \]

is an equivalence. This again follows from (3.12), as desired.

3.3.7

While Theorem 3.2.7 realizes the neutral block of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$, this may be applied to other blocks as follows. Fix a double coset

\[ y \in W_\lambda \backslash \tilde{W} / W_f \]

with associated minimal element $y$ and block $\mathsf {Whit}_{\lambda, y}^{\operatorname {aff}}$.

Proposition 3.3.8 Convolution with the clean object

\[ j_{y, *} \in D_{-\kappa}(I, \lambda \backslash \mathfrak{L} G / I, - y^{{-}1} \cdot \lambda) \]

yields a $t$-exact equivalence

\[ \mathsf{Whit}_{y^{{-}1} \cdot \lambda, e}^{\operatorname{aff}} \xrightarrow{\sim} \mathsf{Whit}_{\lambda, y}^{\operatorname{aff}}. \]

3.4.1

We would like to relate an arbitrary block $\mathsf {Whit}_{\lambda, y}^{\operatorname {aff}}$ of $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ to Kac–Moody representations. Via Proposition 3.3.8, we should apply Theorem 3.2.7 to $\mathsf {Whit}_{y^{-1} \cdot \lambda, e}^{\operatorname {aff}}$. Therefore, we would like to produce a suitable Verma module in

(3.23)\begin{equation} \widehat{\mathfrak{g}}_{-\kappa}{-}\mathsf{mod}^{I, y^{{-}1} \cdot \lambda}. \end{equation}

It will be useful to expand the collection of available Verma modules. For example, if $-\kappa$ is positive rational and the twist $y^{-1} \cdot \lambda$ is trivial, there are no irreducible Verma modules in (3.23). To address this, note that for any integral level $\kappa '$ for $G$ one has a tautological $t$-exact equivalence

\[ D_{-\kappa}(I, y^{{-}1} \cdot \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi) \simeq D_{-\kappa + \kappa'}(I, y^{{-}1} \cdot \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi). \]

We may further increase our supply of integral levels and characters as follows. Write $G_s$ for the simply connected form of $G$ and $I_s$ for the Iwahori subgroup of its loop group. Then the tautological embedding

\[ D_{-\kappa}( I_s, y^{{-}1} \cdot \lambda \backslash \mathfrak{L} G_s / \mathfrak{L} N^-, \psi) \rightarrow D_{-\kappa}(I, y^{{-}1} \cdot \lambda \backslash \mathfrak{L} G / \mathfrak{L} N^-, \psi) \]

induces an equivalence of neutral blocks.

3.4.2

To analyze when we may find the desired Verma modules, we will need some basic properties of the relevant integral Weyl group. So for an arbitrary level $\kappa _{\circ }$ let us denote by $W_{\mathfrak {g}, \kappa _{\circ }}$ the integral Weyl group of $0 \in \mathfrak {t}^*$ at level $\kappa _\circ$, i.e. what was denoted by $W_0$ in the notation of § 3.2.3.

To describe the simple reflections in $W_{\mathfrak {g}, \kappa _{\circ }}$, recall the canonical identification of $W$ with the semidirect product

\[ W_f \ltimes \check{Q}. \]

For any finite coroot $\check {\alpha }$ we denote by $s_{\check {\alpha }}$ the corresponding reflection in $W_f$, and for an element $\check {\lambda }$ of the coroot lattice $\check {Q}$ we write $t^{\check {\lambda }}$ for the corresponding translation in $W$. In addition, let us write $\check {\theta }_s$ for the short dominant coroot, $\check {\theta }_l$ for the long dominant coroot, and $r$ for the lacing number of $\mathfrak {g}$.

Lemma 3.4.3 For any $\kappa _{\circ }$ and integral level $\kappa '$, there are canonical identifications

(3.24)\begin{equation} W_{\mathfrak{g}, \kappa_{{\circ}}} \simeq W_{\mathfrak{g}, -\kappa_{{\circ}}} \simeq W_{\mathfrak{g}, \kappa_{{\circ}} + \kappa'} \end{equation}

intertwining their inclusions into $W$. If $\kappa _{\circ }$ is irrational, then $W_{\mathfrak {g}, \kappa _{\circ }} \simeq W_f$, i.e. they coincide as subgroups of $W$. If $\kappa _{\circ }$ is rational, write it as

\[ \kappa_{{\circ}} = \biggl({-}h^\vee + \frac{p}{q}\biggr)\ \kappa_b, \]

where $h^\vee$ is the dual Coxeter number, $p$ and $q$ are coprime integers, and $\kappa _b$ is the basic level. Then $W_{\mathfrak {g}, \kappa _{\circ }}$ has simple reflections given by the simple reflections of $W_f$ and the additional reflection

(3.25) \begin{equation} s_{0, \kappa_\circ} = \begin{cases} s_{\check{\theta}_s} t^{q \cdot \check{\theta}_s } & {\rm if}\ (q, r) = 1, \\ s_{\check{\theta}_l} t^{q/r \cdot \check{\theta}_l } & {\rm if}\ (q, r) = r. \end{cases} \end{equation}

Proof. Recall the standard enumeration of the affine real coroots as

\[ \check{\Phi} \simeq \check{\Phi}_f \times \mathbb{Z}, \]

where $\check {\Phi }_f$ denotes the finite coroots. With this enumeration, the element $\check {\alpha }_n \in \mathfrak {t} \oplus \mathbb {C} \mathbf {1}$, for $\check {\alpha } \in \check {\Phi }_f$ and $n \in \mathbb {Z}$, is given by

\[ \check{\alpha} + n \frac{\kappa_{{\circ}}(\check{\alpha}, \check{\alpha})}{2} \, \mathbf{1}. \]

In particular, $\check {\alpha }_n$ belongs to $W_{\mathfrak {g}, \kappa _\circ }$ if and only if

\[ \langle \check{\alpha}_n, 0 \rangle = n\frac{\kappa_\circ( \check{\alpha}, \check{\alpha})}{2} \]

is an integer, which straightforwardly implies the claims of the lemma.

3.4.4

Having explicitly identified the Coxeter generators of the integral Weyl group, we will now obtain for most levels highest weights with prescribed stabilizers within it. Let us formulate this problem precisely. If we write $\Lambda _{G_s}$ for the weight lattice and recall that $I_s$ denotes the Iwahori subgroup of $\mathfrak {L} G_s$, then

\[ \widehat{\mathfrak{g}}_{\kappa_{{\circ}}}{-}\mathsf{mod}^{I_s} \]

is compactly generated by the Verma modules $M_\lambda$ for $\lambda \in \Lambda _{G_s}$. In particular, this category has highest weights consisting of the weight lattice. Let us say that a level $\kappa _{\circ }$ is good if for any finite parabolic subgroup $W_\circ$ of $W_{\mathfrak {g}, \kappa _\circ }$ there exists an integral level $\kappa '$ and a simple Verma module

\[ M_\nu \in \widehat{\mathfrak{g}}_{\kappa_{{\circ}} + \kappa'}{-}\mathsf{mod}^{I_s} \]

whose highest weight has stabilizer $W_\circ$. Let us classify the good levels.

Proposition 3.4.5 Every irrational level is good. A rational level

\[ \kappa_\circ{=} \biggl({-}h^\vee + \frac{p}{q}\biggr) \,\kappa_b, \]

where $p$ and $q$ are coprime integers, is good if and only if $q$ is coprime to the number $n(\mathfrak {g})$ associated to $\mathfrak {g}$ in Table 1.

Proof. For $\kappa _{\circ }$ irrational, the claim is clear as $W_{\mathfrak {g}, \kappa _{\circ }} \simeq W_f$. For $\kappa _{\circ }$ rational, which we may take to be negative, a weight $\lambda \in \Lambda _{G_s}$ is antidominant if and only if

\[ \langle \lambda + \rho, \check{\alpha}_i \rangle \leqslant 0 \quad \text{for } i \in {\mathcal I} \quad \text{and} \quad \begin{cases} \langle \lambda + \rho, \check{\theta}_s \rangle \geqslant -p & \text{if } (q,r) = 1,\\ \langle \lambda + \rho, \check{\theta}_l \rangle \geqslant -p & \text{if } (q,r) = r, \end{cases} \]

as follows from (3.25). Let us write $\omega _i,$ with $i \in {\mathcal I}$, for the fundamental weights and write the dominant coroots as sums of simple coroots

\[ \check{\theta}_s = \sum_{i \in {\mathcal I}} n_i \check{\alpha}_i, \quad \check{\theta}_l = \sum_{i \in {\mathcal I}} m_i \check{\alpha}_i \quad \text{for } n_i, m_i \in \mathbb{Z}^{\geqslant 0}. \]

Recall the standard correspondence between finite parabolic subgroups $W_\circ$ of $W_{\mathfrak {g}, \kappa _{\circ }}$ and nonempty faces of the above alcove, which associates to a face the stabilizer of any interior point. It follows that, after the transformation $\lambda \mapsto -\lambda - \rho$, we are looking for points of $\Lambda _{G_s}$ within the alcove with vertices at

\[ 0 \quad \text{and} \quad \begin{cases} \dfrac{p}{n_i} \ \omega_i \quad \text{for } i \in {\mathcal I} & \text{if } (q,r) = 1, \\ \dfrac{p}{m_i} \ \omega_i \quad \text{for } i \in {\mathcal I} & \text{if } (q,r) = r. \end{cases} \]

Recalling that we are free to replace $p$ by any element of $p + q \mathbb {Z}$, it is straightforward to see that we can find points of $\Lambda _{G_s}$ in the interior of every face of the alcove if and only if for each $i \in {\mathcal I}$ one has

\[ \begin{cases} p \in (n_i, q) \quad \text{for } i \in {\mathcal I} & \text{if } (q,r) = 1, \\ p \in (m_i, q)\quad \text{for } i \in {\mathcal I} & \text{if } (q,r) = r. \end{cases} \]

To see this, note that these conditions are tautologically equivalent to being able to realize each vertex of the alcove as a point of $\Lambda _{G_s}$, so they are necessary. To see that they are sufficient, suppose they are satisfied. Via this assumption, for any positive integer $N$ we may replace $p$ with an element of $p + q \mathbb {Z}$ so that

\[ \frac{p}{n_i} \in \mathbb{Z}^{\geqslant N} \quad \text{for all } i \in {\mathcal I}. \]

In particular, we may assume that $N$ is greater than the number of vertices of the alcove, i.e. $N > \lvert I \rvert + 1$. In this case, every face of the alcove contains an interior point expressible as a convex combination of the vertices with coefficients in $({1}/{N})\mathbb {Z}$. As such a convex combination is a point of $\Lambda _{G_s}$, we are done.

Finally, recalling the $n_i$ and $m_i$ for each type (cf. Plates I–IX of [Reference BourbakiBou02]) yields the entries of Table 1. Specifically, if $n_i$ is prime, then $p \in (n_i, q)$ if and only if $(n_i, q) = 1$; and if $n_i$ is composite, then each of its prime divisors occurs as another $n_{i'}$.

3.4.6

In what follows, we will be most concerned with the Whittaker category on $\mathfrak {L} G/I$, i.e. $\mathsf {Whit}_{\lambda }^{\operatorname {aff}}$ for $\lambda = 0$. In this case, we replace $\lambda$ by the level $\kappa$.

In other words, $\mathsf {Whit}_{\kappa }^{\operatorname {aff}} := \mathsf {Whit}_0^{\operatorname {aff}}$. We similarly denote the summands $\mathsf {Whit}_{0,y}^{\operatorname {aff}}$ of this category considered in Theorem 3.2.7 by $\mathsf {Whit}_{\kappa,y}^{\operatorname {aff}}$.

3.4.7

Let us obtain for good levels $\kappa _\circ$ the Kac–Moody realization of blocks of the Whittaker category. Fix a double coset, whose minimal-length element we denote by $y$, in

\[ W_{\mathfrak{g}, \kappa_{{\circ}}} \backslash \tilde{W} / W_f. \]

Proof. By Proposition 3.3.8 and Theorem 3.2.7, it is enough to produce a simple Verma module in

\[ \widehat{\mathfrak{g}}_{\kappa_{{\circ}} + \kappa'}{-}\mathsf{mod}^{I, y^{{-}1} \cdot 0} \]

whose (antidominant) highest weight $\Theta$ has stabilizer under the dot action of $W$ given by

\[ y^{{-}1} W_{\mathfrak{g}, \kappa_{{\circ}}} y \cap W_f. \]

By (3.9), it is equivalent to produce a simple Verma module in

\[ \widehat{\mathfrak{g}}_{\kappa_{{\circ}} + \kappa'}{-}\mathsf{mod}^{I} \]

whose highest weight $y \cdot \Theta$ has stabilizer given by

(3.26)\begin{equation} W_{\mathfrak{g}, \kappa_{{\circ}}} \cap yW_fy^{{-}1}. \end{equation}

The latter is provided by Proposition 3.4.5, as desired.

3.5.1

To relate blocks of Category ${\mathcal O}$ for $\widehat {\mathfrak {g}}_{-\kappa }$ and $\widehat {\check {\mathfrak {g}}}_{\check {\kappa }}$, we would like to use the following result of Fiebig.

Let $\mathfrak {k}$ be an affine Lie algebra with Cartan subalgebra $\mathfrak {h}$. Fix $\alpha \in \mathfrak {h}^*$ such that the Verma module $M_\alpha$ is simple, and write ${\mathcal O}_\alpha$ for the corresponding block of Category ${\mathcal O}$ for $\mathfrak {k}$. Let us write $W_\alpha$ for the integral Weyl group of $\alpha$ and $W_\alpha ^\circ$ for its subgroup stabilizing $\alpha$ under the dot action.

3.5.4

To apply Theorem 3.5.2 in our situation, we must relate $W_{-\kappa, \mathfrak {g}}$ and $W_{\check {\kappa }, \check {\mathfrak {g}}}$.

To do so, fix a level $\kappa _{\circ }$ for $\mathfrak {g}$. Let us write $\check {Q}$ for the coroot lattice and $Q$ for the root lattice of $\mathfrak {g}$. Associated to $\kappa _\circ$ is a map

\[ (\kappa_\circ- {\kappa_{\mathfrak{g},c}}): \check{Q} \rightarrow Q \otimes_{\mathbb{Z}} \mathbb{C}, \]

where $\kappa _{\mathfrak {g}, c}$ denotes the critical level for $\mathfrak {g}$. In particular, we may consider the sublattice of $\check {Q}$ given by

\[ \check{Q}_{\kappa_\circ} := \{ \check{\lambda} \in \check{Q}: (\kappa_\circ- \kappa_{\mathfrak{g}, c}) (\check{\lambda}) \in Q \}. \]

Suppose that $\kappa _\circ$ is noncritical. Recall that if we write $\check {\kappa }_{\check {\mathfrak {g}}, c}$ for the critical level for $\check {\mathfrak {g}}$, then by definition the dual nondegenerate bilinear form of $\kappa _{\circ } - \kappa _{\mathfrak {g}, c}$ is $\check {\kappa }_\circ - \check {\kappa }_{\check {\mathfrak {g}},c}$. It follows that we have a canonical identification

(3.27)\begin{equation} -(\kappa_\circ- \kappa_{\mathfrak{g}, c}): \check{Q}_{\kappa_\circ} \simeq Q_{\check{\kappa}_\circ}: -(\check{\kappa}_\circ- \check{\kappa}_{\check{\mathfrak{g}}, c}). \end{equation}

3.5.5

With this, we may canonically identify the integral Weyl groups on the opposite sides of quantum Langlands duality.

Theorem 3.5.6 For any level $\kappa _\circ$, under the identification $W \simeq W_f \ltimes \check {Q}$ one has

(3.28)\begin{equation} W_{\mathfrak{g}, \kappa_\circ} \simeq W_f \ltimes \check{Q}_{\kappa_\circ}. \end{equation}

Moreover, for a noncritical level $\kappa _\circ$, there is a canonical isomorphism of Coxeter systems

(3.29)\begin{equation} W_{\mathfrak{g}, \kappa_\circ} \simeq W_{\check{\mathfrak{g}}, \check{\kappa}_\circ}. \end{equation}

Proof of Theorem 3.5.6 We begin with (3.28). Recall from the proof of Lemma 3.4.3 that for a finite coroot $\check {\alpha }$ and an integer $n$, the affine coroot $\check {\alpha }_n$ belongs to $W_{\mathfrak {g}, \kappa _\circ }$ if and only if

\[ \langle \check{\alpha}_n, 0 \rangle = n\frac{\kappa_\circ( \check{\alpha}, \check{\alpha})}{2} \]

is an integer. This integrality condition may be rewritten as

\[ n \kappa_\circ(\check{\alpha}) \in \mathbb{Z} \alpha, \]

which in turn is equivalent to $n \check {\alpha } \in \check {Q}_{\kappa _\circ }.$ As the affine reflection in $W$ corresponding to $\check {\alpha }_n$ is explicitly given by $s_{\check {\alpha }} t^{n \check {\alpha }},$ it follows that we have an inclusion

(3.30)\begin{equation} W_{\mathfrak{g}, \kappa_\circ} \subset W_f \ltimes \check{Q}_{\kappa_\circ} \end{equation}

and that the left-hand side includes the translations $t^{n \check {\alpha }}$ for $n \check {\alpha }$ as above. To see that (3.30) is an equality, it suffices to show that $\check {Q}_{\kappa _\circ }$ lies in the left-hand side. But if we write an element $\check {\lambda }$ of $\check {Q}$ as a linear combination

\[ \check{\lambda} = \sum_{i \in {\mathcal I}} n_i \check{\alpha}_i \quad \text{for } i \in {\mathcal I}, \]

we have that $\kappa _\circ (\lambda )$ lies in $Q$ if and only if $\kappa _\circ ( n_i \check {\alpha }_i)$ lies in $Q$ for all $i \in {\mathcal I}$. Considering the affine coroots $(\check {\alpha }_i)_{n_i}$ for $i \in {\mathcal I}$ and composing the translational parts of their reflections yields the desired equality.

Let us use the equality (3.28) to prove (3.29). Via (3.24), we may assume that $\kappa _\circ$ is negative. Under this assumption, we will show that the composite identification

\[ W_{\mathfrak{g}, \kappa_\circ} \overset{(3.28)}{\simeq} W_f \ltimes \check{Q}_{\kappa_\circ} \overset{(3.27)}{\simeq} W_f \ltimes Q_{\check{\kappa}_\circ} \overset{(3.28)}{\simeq} W_{\check{\mathfrak{g}}, \check{\kappa}_\circ} \]

is an isomorphism of Coxeter systems; that is, we claim that the sets of simple reflections are exchanged under the identification

(3.31)\begin{equation} -(\kappa_{{\circ}} - \kappa_{\mathfrak{g}, c}): W_f \ltimes \check{Q}_{\kappa_\circ} {\simeq} \ W_f \ltimes Q_{\check{\kappa}_\circ}. \end{equation}

If $\kappa _\circ$ is irrational, this is clear, as both sides are $W_f$. Otherwise, let us write the level as

(3.32)\begin{equation} \kappa_\circ= \biggl({-}h^\vee+ \frac{p}{q}\biggr)\, \kappa_b, \end{equation}

where $p$ and $q$ are positive coprime integers. Recall the affine simple reflection $s_{0, \kappa _{\circ }}$ from Lemma 3.4.3. Applying (3.31) to it and writing $\theta _s$ for the short dominant root and $\theta _l$ for the long dominant root, we obtain

(3.33)\begin{equation} -(\kappa_\circ- \kappa_{{\mathfrak{g}}, c})\, s_{0, \kappa_\circ} = \begin{cases} s_{\theta_l} t^{p \cdot \theta_l } & \text{if } (q, r) = 1 , \\ s_{\theta_s} t^{p \cdot \theta_s } & \text{if } (q, r) = r. \end{cases} \end{equation}

To finish, recall that $\check {\kappa }_\circ$ is given by

\[ \check{\kappa}_\circ = \biggl({-}h^\vee_{\check{\mathfrak{g}}} + \frac{q}{pr}\biggr) \, \check{\kappa}_{\check{\mathfrak{g}},b}, \]

where $h^\vee _{\check {\mathfrak {g}}}$ and $\check {\kappa }_{\check {\mathfrak {g}},b}$ are the dual Coxeter number and basic level for $\check {\mathfrak {g}}$, respectively. Comparing the analogue of (3.25) for $\check {\mathfrak {g}}$ to (3.33) shows the intertwining of the affine simple reflections by (3.31), as desired.

3.5.8

We may apply these results as follows. Suppose that $\kappa$ and $\check {\kappa }$ are dual levels and that $\kappa '$ is an integral level for $G_s$. Suppose we are given a $y \in \tilde {W}$ of minimal length in $W_{\mathfrak {g}, -\kappa + \kappa '}y$ and a simple Verma module

\[ M_\mu \in \widehat{\mathfrak{g}}_{-\kappa + \kappa'}{-}\mathsf{mod}^{I_s, y^{{-}1} \cdot 0}. \]

Suppose we are further given a simple Verma module $M_{\check {\nu }}$ in $\widehat {\check {\mathfrak {g}}}_{\check {\kappa }}{-}\mathsf{mod}^{\check {I}_s}$, such that the stabilizers of $\mu$ and $\check {\nu }$ are identified via

(3.34)\begin{equation} y^{{-}1}W_{\mathfrak{g}, -\kappa + \kappa'}y \simeq W_{\mathfrak{g}, -\kappa + \kappa'} \overset{(3.24)}{\simeq} W_{\mathfrak{g}, \kappa} \overset{(3.29)}{\simeq} W_{\check{\mathfrak{g}}, \check{\kappa}}. \end{equation}

Corollary 3.5.9 In the above situation, there is a $t$-exact equivalence

\[ \widehat{\mathfrak{g}}_{-\kappa + \kappa'}{-}\mathsf{mod}^{I_s, y^{{-}1} \cdot 0}_\mu \simeq \widehat{\check{\mathfrak{g}}}_{\check{\kappa}}{-}\mathsf{mod}^{\check{I}_s}_{\check{\nu}}. \]

Proof. For either category, which we temporarily denote by ${\mathcal C}$, and the finite-length objects in its heart, which we denote by ${\mathcal C}^{\heartsuit, c}$, the canonical map

\[ D^b( {\mathcal C}^{\heartsuit, c}) \rightarrow {\mathcal C} \]

realizes the latter as the ind-completion of the former. To see this, note that since the blocks contain a simple Verma module, ${\mathcal C}^{\heartsuit, c}$ indeed consists of compact objects, and the fully faithfulness may be checked from Verma to dual Verma modules. Either ${\mathcal C}^{\heartsuit, c}$ is a block of Category ${\mathcal O}$ for the corresponding affine algebra, whence we are done by the assumptions on $\mu$ and $\check {\nu }$, the identification of Coxeter systems (3.34), and Theorem 3.5.2.

3.7.1

Recall the canonical bijection between the simple roots of $\mathfrak {g}$ and $\check {\mathfrak {g}}$, which were indexed by ${\mathcal I}$. In particular, to a standard parahoric subgroup

\[ P \subset \mathfrak{L} G, \]

which corresponds to a subset ${\mathcal J}$ of ${\mathcal I}$, we may associate a dual parahoric

\[ \check{P} \subset \mathfrak{L}\check{G}. \]

3.7.2

Let us obtain a parahoric extension of the tamely ramified fundamental local equivalence. In particular, we continue to assume that $G$ is simple of adjoint type.

Theorem 3.7.3 For a good, negative level $\kappa$, there is a $t$-exact equivalence

(3.40)\begin{equation} D_{\kappa}( \mathfrak{L} N, \psi \backslash \mathfrak{L} G / P) \simeq \widehat{\check{\mathfrak{g}}}_{\check{\kappa}}{-}\mathsf{mod}^{\check{P}}. \end{equation}

Proof. It is enough to produce an equivalence, which we denote provisionally by a dotted line, fitting into the commutative diagram

(3.41)

where we normalize the pull-back $\pi ^{!*}$ associated to $\pi : \mathfrak {L} G / I \rightarrow \mathfrak {L} G/ P$ to be $t$-exact. Here, as in the proof of Corollary 3.5.9, the superscripts $\heartsuit$ and $c$ denote compact objects in the heart, which in the present situation are equivalent to finite-length objects in the heart. Noting that both vertical arrows in (

3.41

) are full embeddings of Serre subcategories, it is enough to show that the essential images of the simple objects under are intertwined by (3.35). To see this claim, recall that we denoted the subset of simple roots corresponding to the dual parahorics $P$ and $\check {P}$ by ${\mathcal J}$. With this, the simple objects on the Whittaker side lying in the essential image are intermediate extensions from the orbits

\[ Ix\mathfrak{L} N^- \quad \text{for } x \in \tilde{W}, \]

where $x$ satisfies the three conditions

(3.42)\begin{equation} s_j x < x \;\; \forall\, j \in {\mathcal J}, \quad xs_i < x \;\; \forall \, i \in {\mathcal I}, \quad \text{and} \quad W_{\mathcal J} x s_i < W_{\mathcal J} x \;\; \forall\, i \in {\mathcal I}. \end{equation}

The simple objects on the Kac–Moody side lying in the essential image have highest weights $\check {\lambda }$ satisfying

(3.43)\begin{equation} s_j \cdot \check{\lambda} < \check{\lambda} \quad \forall\, j \in {\mathcal J}. \end{equation}

Write $\check {\lambda } = x \cdot -\check {\rho }$ for $x \in \tilde {W}$ acting as in (3.37). We may assume that $x$ is of maximal length in its left $W_f$ coset, in which case (3.43) is equivalent to $x$ satisfying the three conditions

(3.44)\begin{equation} s_j x < x\;\; \forall\, j \in {\mathcal J}, \quad xs_i < x\;\; \forall\, i \in {\mathcal I}, \quad \text{and} \quad s_j x W_f < x W_f\;\; \forall\, j \in {\mathcal J}. \end{equation}

We finish by noting that (3.42) and (3.44) describe the same subset of $\tilde {W}$, namely the unique elements of maximal length in double cosets $W_{\mathcal J} x W_f$ such that the double coset satisfies

\[ x( \check{\Phi}_f) \cap \check{\Phi}_{{\mathcal J}} = \emptyset, \]

where $\check {\Phi }_f$ denotes the finite coroots and $\check {\Phi }_{{\mathcal J}}$ the coroots of the Levi associated to ${\mathcal J}$.

3.7.4

We finish with two remarks.

Remark 3.7.5 Applying Theorem 3.7.3 in the maximal case, i.e. for the parahorics given by the arc groups $\mathfrak {L}^+ G$ and $\mathfrak {L}^+ \check {G}$, we obtain the spherical fundamental local equivalence for good levels, namely

(3.45)\begin{equation} \mathsf{Whit}_{\kappa}^{\operatorname{sph}} := D_\kappa( \mathfrak{L} N, \psi \backslash \mathfrak{L} G / \mathfrak{L} ^+ G) \simeq \widehat{\check{\mathfrak{g}}}_{\check{\kappa}}{-}\mathsf{mod}^{\mathfrak{L}^+ \check{G}}. \end{equation}

One can reasonably ask whether the methods of this paper can be directly applied to obtain this without first proving Theorem 3.6.1. The arguments indeed apply mutatis mutandis, where one works throughout with double cosets of $W_{\mathfrak {g}, \kappa }$ rather than one-sided cosets, exactly as in the proof of Theorem 3.7.3.

In this approach, to cross Langlands duality one needs the parabolic variant of Fiebig's theorem, Theorem 3.5.2. This is indeed true, and may be deduced from the usual case by an argument similar to that for Theorem 3.7.3.