2. Notation

Let $\mathcal{O}$ be a discrete valuation ring with residue field $\mathbb{F}$ and fraction field E. Let G be a split connected reductive group over $\operatorname{Spec} \mathcal{O}$ with connected fibres together with a choice of maximal torus T. Let $X_*(T) = \operatorname{Hom}(\mathbb{G}_m,T)$ and $X^*(T) = \operatorname{Hom}(T,\mathbb{G}_m)$ and write $\langle ~,~\rangle$ for the natural pairing $X^*(T) \times X_*(T) \rightarrow \mathbb{Z}$ . Let $R^\vee \subset X^*(T)$ be the roots (G,T) and for $\alpha^\vee \in R^\vee$ write $\alpha \in X_*(T)$ for the corresponding coroot. Let W be the Weyl group of (G,T). Choose a set of positive roots $R_+^\vee \subset R^\vee$ , with associated Borel B. Let $R_+$ denote the corresponding set of positive coroots. Recall

$$ \lambda^\vee \leq \mu^\vee \Leftrightarrow \mu^\vee - \lambda^\vee \in \sum_{\alpha \in R_+^\vee} \mathbb{Z}_{\geq 0} \alpha^\vee, $$

and that $\lambda^\vee \in X^*(T)$ is dominant if $\langle \lambda^\vee, \alpha\rangle \geq 0$ for all positive coroots $\alpha \in R_+$ . We say $\lambda^\vee$ is strictly dominant if $\langle \lambda^\vee, \alpha \rangle \geq 1$ . We likewise make sense of $\leq$ on $X_*(T)$ as well as dominant and strictly dominant $\lambda \in X_*(T)$ .

Notice that if $\rho \in X_*(T)$ is a twisting element then $\rho - \frac{1}{2}\sum_{\alpha \in R_+} \alpha$ is W-invariant. Also $\lambda \in X_*(T)$ is strictly dominant if and only if $\lambda -\rho$ is dominant.

3. Torsors

Throughout this paper we view G-torsors from the following two equivalent viewpoints.

• – A G-torsor $\mathcal{E}$ on $\operatorname{Spec}A$ is an A-scheme equipped with an action of G so that fppf (equivalently etale) locally on A one has $\mathcal{E} \cong G \times_{\mathcal{O}} \operatorname{Spec}A$ .

• – A fibre functor (a faithful exact tensor functor) from the category of representations of G on finite free $\mathcal{O}$ -modules into the category of projective A-modules.

Any G-torsor in the first sense induces a fibre functor which sends a representation $\chi: G \rightarrow \operatorname{GL}(V)$ onto the contracted product

$$\mathcal{E}^\chi := \mathcal{E} \times^\chi V = \mathcal{E} \times V / \sim.$$

That this construction produces an equivalence of categories is proved in, for example, [Reference BroshiBro13, 4.8]. We always write $\mathcal{E}^0$ for the trivial G-torsor and a trivialisation of a G-torsor on $\operatorname{Spec}A$ is an isomorphism $\mathcal{E} \cong \mathcal{E}^0$ over $\operatorname{Spec}A$ .

4. Affine Grassmannians

Fix an integer $e \geq 1$ and pairwise distinct $\pi_1,\ldots,\pi_e$ in the maximal ideal of $\mathcal{O}$ . For any $\mathcal{O}$ -algebra A we write $E(u) = \prod_{i=1}^e (u-\pi_i) \in A[u]$ .

• – $\mathcal{E}$ is a G-torsor over $\operatorname{Spec}A[u]$ ;

• – $\iota$ is a trivialisation of $\mathcal{E}$ over the open subscheme $\operatorname{Spec}A[u,E(u)^{-1}]$ , i.e. an isomorphism

  $$ \mathcal{E}|_{\operatorname{Spec}A[u,E(u)^{-1}]} \cong \mathcal{E}^0|_{\operatorname{Spec}A[u,E(u)^{-1}],} $$
  where $\mathcal{E}^0$ denotes the trivial G-torsor.

We also consider variants $\operatorname{Gr}_{G,i}$ of $\operatorname{Gr}_G$ for $i =1,\ldots,e$ which are again projective ind-schemes over $\mathcal{O}$ , and whose A-points classify isomorphism classes of pairs $(\mathcal{E},\iota)$ with $\mathcal{E}$ a G-torsor on $\operatorname{Spec}A[u]$ and $\iota$ is a trivialisation of $\mathcal{E}$ over the open subscheme $\operatorname{Spec}A[u,(u-\pi_i)^{-1}]$ . Notice that for each i there are natural closed immersions $\operatorname{Gr}_{G,i} \rightarrow \operatorname{Gr}_G$ . Each of $\operatorname{Gr}_G$ and $\operatorname{Gr}_{G,i}$ are also functorial in G.

Remark 4.2. When $G = \operatorname{GL}_n$ the above functor is a colimit over $a \geq 0$ of the functors sending an $\mathcal{O}$ -algebra A onto the set of rank n projective A[u] submodules

$$E(u)^{a} A[u]^n \subset \mathcal{E} \subset E(u)^{-a} A[u]^n.$$

Since a submodule $\mathcal{E} \subset E(u)^a A[u]^n$ is A[u]-projective of rank n if and only if $E(u)^aA[u]^n /\mathcal{E}$ is A-projective (see [Reference ZhuZhu17, Lemma 1.1.5]) each subfunctor is represented by a subfunctor of the Grassmannian classifying projective A-submodules of $E(u)^{-a}A[u]^n/E(u)^aA[u]^n$ , which shows the ind-representability of $\operatorname{Gr}_{\operatorname{GL}_n}$ .

For general G one chooses a faithful representation into $\operatorname{GL}_n$ and, using [Reference ZhuZhu17, 1.2.6], identifies $\operatorname{Gr}_G$ as a closed sub-ind-scheme of $\operatorname{Gr}_{\operatorname{GL}_n}$ .

If A is an E-algebra and $n_i \in \mathbb{Z}_{\geq 0}$ then, since E[u] is principal ideal domain, the product of the quotient maps describes an isomorphism

$$\frac{A[u]}{\prod_{i=1}^e (u-\pi_i)^{n_i}} \cong \prod_{i=1}^e \frac{A[u]}{(u-\pi_i)^{n_i}}.$$

In particular, this gives an isomorphism $\widehat{A[u]}_{E(u)}\cong \prod_{i=1}^e \widehat{A[u]}_{(u-\pi_i)}$ where the completions are respectively taken against the ideals generated by E(u) and $(u-\pi_i)$ . As a consequence, we obtain the following.

Corollary 4.4. There is an isomorphism

$$ \operatorname{Gr}_G \otimes_{\mathcal{O}} E \cong (\operatorname{Gr}_{G,1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} \operatorname{Gr}_{G,e}) \otimes_{\mathcal{O}} E, $$

written $(\mathcal{E},\iota) \mapsto (\mathcal{E}_i,\iota_i)_{i=1,\ldots,e}$ on A-valued points, so that

$$ \mathcal{E} \otimes_{A[u]} \widehat{A[u]}_{E(u)} = \prod_{i=1}^e \mathcal{E}_i \otimes_{A[u]} \widehat{A[u]}_{u-\pi_i}, $$

with $\iota = \prod_i \iota_i$ .

The isomorphism in Corollary 4.4 has an alternative description. For any $\mathcal{O}$ -algebra A consider the open subsets

$$ U_i = \operatorname{Spec}A\bigg[u,\prod_{j \neq i} (u-\pi_j)^{-1}\bigg], \quad V_i = \operatorname{Spec}A[u,(u-\pi_i)^{-1}] $$

of $\operatorname{Spec}A[u]$ . Then $V_i = \bigcup_{j \neq i} U_j$ , $V_i \cap U_i = \operatorname{Spec}A[u,E(u)^{-1}]$ and, if A is an E-algebra, then

$$ \operatorname{Spec}A[u] = \bigcup_i U_i = V_i \cup U_i. $$

Then $\mathcal{E}_i$ is the G-torsor obtained by glueing $\mathcal{E}|_{U_i}$ and $\mathcal{E}^0|_{V_i}$ along the restriction of $\iota$ to $U_i \cap V_i= \operatorname{Spec}A[u,E(u)^{-1}]$

Notation 4.5. For $\lambda \in X_*(T)$ write $\mathcal{E}_{\lambda,i}$ for the $\mathcal{O}$ -valued point

$$ (\mathcal{E}^0,(u-\pi_i)^\lambda) \in \operatorname{Gr}_{G,i}, $$

where $\mathcal{E}^0$ denotes the trivial G-torsor on $\operatorname{Spec}\mathcal{O}[u]$ and $(u-\pi_i)^{\lambda}$ denotes the automorphism of $\mathcal{E}^0|_{\operatorname{Spec}\mathcal{O}[u,E(u)^{-1}]}$ induced by multiplication by $\lambda(u-\pi_i) \in G(\mathcal{O}[u,E(u)^{-1}])$ . We then define the locally closed subscheme

$$ \operatorname{Gr}_{G,i,\lambda} \subset \operatorname{Gr}_{G,i} $$

as the orbit of $\mathcal{E}_{\lambda,i}$ under the action of the group scheme $L^+G$ with A-valued points G(A[u]).

Lemma 4.6. For each $\lambda \in X_*(T)$ and $i=1,\ldots,e$ the morphism $G \rightarrow \operatorname{Gr}_{G,i}$ given by $g \mapsto g \mathcal{E}_{\lambda,i}$ induces a closed immersion

$$ G/P_\lambda \rightarrow \operatorname{Gr}_{G,i}, $$

where

$$ P_\lambda = \lbrace g \in G \mid \operatorname{lim}_{t \rightarrow 0} \lambda(t)^{-1} g \lambda(t) \text{ exists} \rbrace. $$

Equivalently, $P_\lambda$ is the parabolic subgroup of G generated by T and the roots subgroups $U_{\alpha^\vee}$ for $\alpha^\vee \in R^\vee$ with $\langle \alpha^\vee ,\lambda \rangle \leq 0$ .

Definition 4.7. For $\mu = (\mu_1,\ldots,\mu_e)$ with $\mu_i \in X_*(T)$ set $M_\mu \subset \operatorname{Gr}_G$ equal to the scheme theoretic image of the composite

$$ ( G/P_{\mu_1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} G/P_{\mu_e}) \otimes_{\mathcal{O}} E \rightarrow ( \operatorname{Gr}_{G,1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} \operatorname{Gr}_{G,e} ) \otimes_{\mathcal{O}} E \cong \operatorname{Gr}_G \otimes_{\mathcal{O}} E \rightarrow \operatorname{Gr}_{G} $$

(the isomorphism coming from Lemma 4.4).

The A-valued points of $G/P_\lambda$ classify filtrations of type $\lambda$ on the trivial G-torsor over $\operatorname{Spec}A$ , i.e. exact tensor functors from the category of representations of G on finite free $\mathcal{O}$ -modules V into the category of filtrations on V by A-modules with projective graded pieces. From this point of view, we have the following.

• – The closed immersion $G/P_\lambda \rightarrow \operatorname{Gr}_{G,i}$ from Lemma 4.6 sends a filtration $\operatorname{Fil}^\bullet$ onto the G-torsor $\operatorname{Fil}^0( G \otimes_A \widehat{A[u]}_{(u-\pi)})$ over $\widehat{A[u]}_{(u-\pi)}$ which, when interpreted as an exact tensor functor on the category of representations of G, is given by

  $$ V \mapsto \sum_{n \in \mathbb{Z}} \operatorname{Fil}^n(V) \otimes_A (u-\pi_i)^{-n} \widehat{A[u]}_{(u-\pi_i)}. $$
  This G-torsor is interpreted as a point of $\operatorname{Gr}_{G,i}$ via the natural trivialisation coming from the identification of $\operatorname{Fil}^0( G \otimes_A \widehat{A[u]}_{(u-\pi_i)})$ after inverting $(u-\pi_i)$ with the base change of the trivial G-torsor underlying $\operatorname{Fil}^\bullet$ to $\widehat{A[u]}_{(u-\pi_i)}[{1}/{(u-\pi_i)}]$ . To verify this assertion it suffices, by functoriality of $G/P_\lambda \rightarrow \operatorname{Gr}_{G,i}$ in G, to consider the case $G = \operatorname{GL}_n$ .

• – Similarly, the closed immersion $( G/P_{\mu_1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} G/P_{\mu_e} ) \rightarrow \operatorname{Gr}_G$ from Definition 4.7 sends an A-valued point corresponding to an e-tuple of filtrations $(\operatorname{Fil}^\bullet_i)_i$ onto the G-torsor $\operatorname{Fil}^0( G \otimes_A \widehat{A[u]}_{E(u)})$ which, from the Tannakian viewpoint, we view as the functor on representations of G given by

  $$ V \mapsto \bigg( \sum_{n \in \mathbb{Z}} \operatorname{Fil}^n(V) \otimes_A (u-\pi_i)^{-n} \widehat{A[u]}_{(u-\pi_i)} \bigg)_{i=1,\ldots,e }. $$
  The target is viewed as a projective $\widehat{A[u]}_{E(u)}$ -module via the isomorphism from 4.1.

5. Various Schubert varieties

Here we introduce some variants on the usual notation of Schubert varieties inside the affine Grassmannian.

Notation 5.1. For $\lambda \in X_*(T)$ write $\mathcal{E}_{\lambda,\mathbb{F}} \in \operatorname{Gr}_G(\mathbb{F})$ for the point corresponding to $(\mathcal{E}^0,u^\lambda)$ . Thus

$$ \mathcal{E}_{\lambda,\mathbb{F}} = \mathcal{E}_{\lambda,i} \otimes_{\mathcal{O}} \mathbb{F}, $$

with $\mathcal{E}_{\lambda,i}$ as defined in Notation 4.5 and for any $i=1,\ldots,e$ . We also write $\operatorname{Gr}_{G,\lambda,\mathbb{F}} = \operatorname{Gr}_{G,\lambda,i} \otimes_{\mathcal{O}} \mathbb{F}$ for any $i=1,\ldots,e$ . Equivalently, $\operatorname{Gr}_{G,\lambda,\mathbb{F}}$ is the $L^+G$ -orbit of $\mathcal{E}_{\lambda,\mathbb{F}}$ .

For $\lambda \in X_*(T)$ the Schubert variety $\operatorname{Gr}_{G,\leq \lambda,\mathbb{F}} \subset \operatorname{Gr}_G \otimes_{\mathcal{O}} \mathbb{F}$ is usually defined as the closure of $\operatorname{Gr}_{G,\lambda,i} \otimes_{\mathcal{O}} \mathbb{F}$ (for any $i=1,\ldots,e)$ . Then $\operatorname{Gr}_{G,\leq \lambda,\mathbb{F}}$ is reduced, irreducible, and can be expressed as

$$ \operatorname{Gr}_{G,\leq \lambda,\mathbb{F}} = \bigcup_{\lambda' \leq \lambda} \operatorname{Gr}_{G,\lambda',\mathbb{F}}\!. $$

We would like to use $\operatorname{Gr}_{G,\leq \mu_1+\cdots+\mu_e,\mathbb{F}}$ as an ambient space in which to study $M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ . However, the containment of $M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ in $\operatorname{Gr}_{G,\leq \mu_1+\cdots+\mu_e,\mathbb{F}}$ is not immediate due to both varieties construction as a closure. This issue can be addressed using the identifications from [Reference Pappas and ZhuPZ13, Reference LevinLev16] of $\operatorname{Gr}_{G,\leq \lambda,\mathbb{F}}$ with the special fibre of mixed characteristic Schubert varieties. However, we instead use give a more simple minded approach and replace $\operatorname{Gr}_{G,\leq \lambda,\mathbb{F}}$ with a moduli construction which is close to (and conjectured to equal) $\operatorname{Gr}_{G,\leq \lambda,\mathbb{F}}$ .

Definition 5.2. Let V be a free $\mathcal{O}$ -module. Any A-valued point of $\operatorname{Gr}_{\operatorname{GL}(V)}$ corresponds to a projective A[u]-submodule of $V \otimes_{\mathcal{O}} A[u,E(u)^{-1}]$ and so, for any e-tuple of integers $(n_i)$ , we may consider the closed subfunctor $Y_{\operatorname{GL}(V)}^{\geq (n_i)} \subset \operatorname{Gr}_{\operatorname{GL}(V)}$ consisting of those $\mathcal{E}$ for which

(5.3) \begin{equation} \mathcal{E} \subset \prod_{i=1}^e (u-\pi_i)^{n_i} V \otimes_{\mathcal{O}} A[u]. \end{equation}

If $\mu = (\mu_1,\ldots,\mu_e)$ with $\mu_i \in X_*(T)$ dominant then define

$$ Y_{G,\leq \mu} = \bigcap_\chi ( \operatorname{Gr}_G \times_{\chi,\operatorname{Gr}_{\operatorname{GL}(V)}} Y_{\operatorname{GL}(V)}^{\geq (\langle w_0\chi^\vee,\mu_i\rangle)} ), $$

where $w_0 \in W$ is the longest element and the intersection runs over irreducible algebraic representations $\chi: G \rightarrow \operatorname{GL}(V)$ of highest weight $\chi^\vee$ . Notice that each $Y_{\operatorname{GL}(V)}^{\geq (n_i)}$ is stable under the action of $L^+\operatorname{GL}(V)$ on $\operatorname{Gr}_{\operatorname{GL}(V)}$ and so $Y_{G,\leq \mu}$ is stable under the $L^+G$ -action on $\operatorname{Gr}_G$ .

Example 5.4. Suppose that $G = \operatorname{GL}_n$ so that A-valued points of $\operatorname{Gr}_G $ identify with projective A[u]-submodules $\mathcal{E} \subset A[u,E(u)^{-1}]^n$ . If

$$\mu = (\mu_1,\ldots,\mu_e), \quad \mu_i = (\mu_{i,1} \geq \cdots \geq \mu_{i,n}),$$

and $\mathcal{E} \in Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F}$ then

(5.5) \begin{equation} \bigwedge^j(\mathcal{E}) \subset \prod_{i=1}^e (u-\pi_i)^{ \mu_{i,n} + \cdots + \mu_{i,n-j+1}} A[u]^{\binom{n}{j}}\end{equation}

for all $j = 1,\ldots,n$ . Indeed, if $\chi:G \rightarrow \operatorname{GL}(V)$ equals the jth exterior power of the standard representation, then the induced morphism $\operatorname{Gr}_G \rightarrow \operatorname{Gr}_{\operatorname{GL}(V)}$ sends $\mathcal{E}$ onto $\bigwedge^j(\mathcal{E})$ and so

$$\operatorname{Gr}_G \times_{\chi,\operatorname{Gr}_{\operatorname{GL}(V)}} Y_{\operatorname{GL}(V)}^{\geq (\langle w_0\chi^\vee,\mu_i\rangle)},$$

is the closed subscheme consisting of $\mathcal{E}$ as in (5.5). This is because

$$\chi^\vee = (\underbrace{1,\ldots,1}_{j\text{ ones}},0,\ldots,0),$$

and so $\langle w_0 \chi^\vee, \mu_i \rangle = \mu_{i,n}+\cdots+\mu_{i,n-j+1}$ . In fact, since every highest weight representation of $\operatorname{GL}_n$ is a quotient of tensor products of these exterior powers representations, conditions (5.5) suffice to determine $Y_{G,\leq \mu}$ .

The following lemma describes the basic properties of $Y_{G,\leq \mu}$ that we need.

Lemma 5.6. Let $\mu_1,\ldots,\mu_e \in X_*(T)$ be dominant. Then $Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F}$ only depends upon $\mu_1+\cdots+\mu_e$ , contains $\operatorname{Gr}_{G,\mu_1+\cdots+\mu_e,\mathbb{F}}$ as an open subset, and

$$ (Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F})_{\operatorname{red}} = \bigcup_{\lambda \leq \mu_1+\cdots+\mu_e} \operatorname{Gr}_{G,\lambda,\mathbb{F}}\!. $$

Proof. That $Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F}$ only depends upon $\mu_1+\cdots+\mu_e$ is clear from the definition. It is also clear from the definition that $Y_{G,\leq \mu}$ contains $\mathcal{E}_{\lambda,\mathbb{F}} \in \operatorname{Gr}_{G}(\mathbb{F})$ if and only if $\lambda \leq \mu_1+\cdots+\mu_e$ . Since $Y_{G,\leq \mu}$ is $L^+G$ -stable it follows that

$$ (Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F})_{\operatorname{red}} = \bigcup_{\lambda \leq \mu_1+\cdots+\mu_e} \operatorname{Gr}_{G,\lambda,\mathbb{F}}\!. $$

It remains to show $\operatorname{Gr}_{G, \mu_1+\cdots{\kern-.5pt},\mu_e,\mathbb{F}}$ is open in $Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F}$ . This will follow if $Y_{G,\leq \mu} \otimes_{\mathcal{O}}\mathbb{F}$ is reduced at $\mathcal{E}_{\mu_1+\cdots+\mu_e,\mathbb{F}}$ , which can be achieved by a simple tangent space computation identifying the tangent space of $Y_{G,\leq \mu}\otimes_{\mathcal{O}} \mathbb{F}$ at $\mathcal{E}_{\mu_1+\cdots+\mu_e,\mathbb{F}}$ with the tangent space of $\operatorname{Gr}_{G,\mu_1+\cdots+\mu_e,\mathbb{F}}$ . See [Reference Kamnitzer, Muthiah and WeekesKMW18, § 3].

Lemma 5.8. Under the isomorphism $\operatorname{Gr}_G \otimes_{\mathcal{O}} E \cong ( \operatorname{Gr}_{G,1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} \operatorname{Gr}_{G,e} ) \otimes_{\mathcal{O}} E$ from Lemma 4.4 we have

$$ Y_{G,\leq \mu} \otimes_{\mathcal{O}} E \cong ( Y_{G,1,\leq \mu_1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} Y_{G,e,\leq \mu_e} ) \otimes_{\mathcal{O}} E, $$

where $Y_{G,i,\leq \mu_i} \subset \operatorname{Gr}_{G,i}$ is defined as a special case of Definition 5.2.

Proof. The lemma reduces to showing that, for any tuple $(n_i)$ of integers and any finite free $\mathcal{O}$ -module V,

$$ Y_{\operatorname{GL}(V)}^{\geq (n_i)} \otimes_{\mathcal{O}} E = ( Y_{\operatorname{GL}(V),1}^{\geq n_1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} Y_{\operatorname{GL}(V),e}^{\geq n_e}) \otimes_{\mathcal{O}} E, $$

under the identification from Lemma 7. This is clear since if A is an E-algebra then the isomorphism $\widehat{A[u]}_{E(u)} \cong \prod_{i=1}^e \widehat{A[u]}_{u-\pi_i}$ identifies the ideal generated by $\prod (u-\pi_i)^{n_i}$ with the product of the ideals generated by $(u-\pi_i)^{n_i}$ .

The reason we introduce Definition 4.4 is because it easily allows us to prove the following.

6. Approximations via $\operatorname{Gr}_G^\nabla$

When $G = \operatorname{GL}_n$ the spaces $M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ were accessed in [Reference BartlettBar24, § 7] by constructing a subfunctor $\operatorname{Gr}^\nabla_G \subset \operatorname{Gr}_G$ with $M_\mu \subset \operatorname{Gr}_G^\nabla$ . The subfunctor $\operatorname{Gr}_G^\nabla$ consists of $\mathcal{E} \in \operatorname{Gr}_G(A)$ which, when viewed as projective A[u]-submodules of $A[u,E(u)^{-1}]^n$ , satisfy

$$E(u)\nabla(\mathcal{E}) \subset \mathcal{E}$$

for $\nabla$ the operator on $A[u,E(u)^{-1}]^n$ given coordinate-wise via ${d}/{du}$ . If $\mathcal{E}$ is generated by $(e_1,\ldots,e_n)X$ for a matrix $X \in\operatorname{GL}_n(A[u,E(u)^{-1}])$ then this is equivalent to asking that

$$E(u)X^{-1}\frac{d}{du}(X) \in \operatorname{Mat}(A[u]).$$

In this section we show how to extend this construction to general G.

The following construction works when $G = \operatorname{Spec}\mathcal{O}_G$ is any affine algebraic group over $\mathcal{O}$ . Set $\mathfrak{g} = \operatorname{Lie}(G)$ . In what follows we will interpret elements of $\mathfrak{g}$ as derivations $\mathcal{O}_G \rightarrow \mathcal{O}$ over $\mathcal{O}$ where $\mathcal{O}_G$ acts on $\mathcal{O}$ via the counit map $e:\mathcal{O}_G \rightarrow \mathcal{O}$ . The logarithmic derivative can then be described as a map

$$ \operatorname{dlog}: G(B) \rightarrow \mathfrak{g} \otimes_{\mathbb{Z}} \Omega_{B/\mathcal{O}} $$

for any ring B. To define this map identify $\Omega_{G/\mathcal{O}} = \mathcal{L}(G)^\vee \otimes_{\mathcal{O}} \mathcal{O}_G$ where $\mathcal{L}(G)$ denotes the translation invariant derivations $\mathcal{O}_G \rightarrow \mathcal{O}_G$ . Then

$$ \mathfrak{g} \otimes_{\mathcal{O}} \Omega_{G/\mathcal{O}} = \operatorname{Hom}_{\mathcal{O}}(\mathcal{L}(G),\mathfrak{g}) \otimes_{\mathcal{O}} \mathcal{O}_G $$

and the map $\mathcal{L}(G) \rightarrow \mathfrak{g}$ given by composition with the counit e defines a canonical global section. For any $g \in G(B)$ define $\operatorname{dlog}(g)$ as the image of this section under $\mathfrak{g} \otimes_{\mathcal{O}} g^*\Omega_{G/\mathcal{O}} \rightarrow \mathfrak{g} \otimes_{\mathcal{O}} \Omega_{B/\mathcal{O}}$ . If A is any $\mathcal{O}$ -algebra and $B = A[u,E(u)^{-1}]$ then we obtain an element

$$ \operatorname{dlog}_u(g) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u,E(u)^{-1}], $$

by evaluating $\operatorname{dlog}(g)$ at the derivation

$$\frac{d}{du}: A[u,E(u)^{-1}] \rightarrow A[u,E(u)^{-1}].$$

This construction is functorial in G.

The following example motivates us calling this construction the logarithmic derivative.

Example 6.1. Let $G = \operatorname{GL}_n$ with coordinates $T_{ij}$ and write $\iota: \mathcal{O}_G \rightarrow \mathcal{O}_G$ for the coinverse map. Write ${d}/{d T_{ij}}$ for the element of $\mathfrak{g}$ sending $T_{ij} \mapsto 1$ and zero on all other coordinates. We claim that the section

(6.2) \begin{equation} \sum_{ij} \frac{d}{dT_{ij}} \otimes \bigg( \sum_m \iota(T_{im}) d(T_{mj}) \bigg) \in \mathfrak{g} \otimes_{\mathcal{O}} \Omega_{G/\mathcal{O}}, \end{equation}

coincides with the map $\mathcal{L}(G) \rightarrow \mathfrak{g}$ given by composition with the counit. If $\Delta: T_{ij} \mapsto \sum_l T_{il} \otimes T_{lj}$ is the comultiplication map then $\mathcal{L}(G)\rightarrow \mathfrak{g}$ has an inverse given by $d \mapsto (\operatorname{id} \otimes d) \circ \Delta$ (see, for example, [Reference MilneMil17, 12.24]). Therefore we can check the claim by evaluating $\sum_m \iota(T_{im}) d(T_{mj})$ at $(\operatorname{id} \otimes ({d}/{d T_{lk}})) \circ \Delta$ . Since

$$ \bigg(\operatorname{id} \otimes \frac{d}{d T_{lk}}\bigg) \circ \Delta: T_{mj} \mapsto \begin{cases} T_{ml} & \!\text{if $j=k$,} \\ 0 & \!\text{otherwise,} \end{cases} $$

this evaluation is equal to

$$ \begin{cases} \sum_m \iota(T_{im}) T_{ml} & \text{ if $j=k$,} \\ 0 & \text{ if $j \neq k$.} \end{cases}$$

Since composing $(\iota \otimes \operatorname{id}) \circ \Delta$ with multiplication $\mathcal{O}_G \otimes \mathcal{O}_G \rightarrow \mathcal{O}_G$ equals the counit e it follows that the above evaluation is 1 if $ij = lk$ and zero otherwise. This verifies our claim. If $g = (g_{ij}) \in G(B)$ has inverse $g^{-1} = (h_{ij})$ then the image of (6.2) in $\mathfrak{g} \otimes \Omega_{B/\mathcal{O}}$ is

$$ \operatorname{dlog}(g) = \sum_{ij} \frac{d}{d T_{ij}} \otimes \bigg( \sum_m h_{im} d(g_{mj}) \bigg). $$

If $B = A[u,E(u)^{-1}]$ and we identify $\mathfrak{g} = \operatorname{Mat}_{n\times n}(\mathcal{O})$ via ${d}/{d T_{ij}}$ then evaluating $\operatorname{dlog}(g)$ at ${d}/{du}$ yields $\operatorname{dlog}_u(g) = g^{-1}{d(g)}/{du}$ .

Remark 6.3 If G is a flat and finite type over $\mathcal{O}$ then $\operatorname{dlog}_u$ can alternatively be constructed using the Tannakian viewpoint. As explained in, for example, [Reference LevinLev15], an element of $\mathfrak{g} \otimes_{\mathcal{O}} A[u]$ is equivalent to a collection of endomorphisms $X_V$ for all representations $G \rightarrow \operatorname{GL}(V)$ of G on finite free $\mathcal{O}$ -modules which are compatible with exact sequences and satisfy $X_{V_1\otimes V_2} = X_{V_1} \otimes 1 + 1 \otimes X_{V_2}$ . Example 6.1 shows that $\operatorname{dlog}_u(g)$ corresponds to the rule sending a representation $\rho$ onto

$$ \rho(g)^{-1} \frac{d}{du}(\rho(g)) \in \operatorname{End}(V) $$

(the compatibility of this rule with exact sequences and tensor products being an easy computation).

Example 6.4. If $G = \mathbb{G}_a$ with coordinate T then write ${d}/{dT}$ for the element of $\mathfrak{g}$ sending $T \mapsto 1$ . In this case the section

$$ \frac{d}{dT} \otimes d(T) \in \mathfrak{g} \otimes_{\mathcal{O}} \Omega_{G/\mathcal{O}} $$

coincides with the map $\mathcal{L}(G) \rightarrow \mathfrak{g}$ given by composition with the counit and so if $g \in \mathbb{G}_a(B)$ then

$$ \operatorname{dlog}(g) = \frac{d}{dT} \otimes d(b). $$

If $B = A[u,E(u)^{-1}]$ and we identify $\mathfrak{g} = \mathcal{O}$ via ${d}/{dT}$ it follows that $\operatorname{dlog}_u(g) = {d(g)}/{du}$ .

The next lemma contains what we will need to compute with $\operatorname{dlog}_u(-)$ .

Definition 6.6. Define $\operatorname{Gr}^{\nabla}_G \subset \operatorname{Gr}_G$ to be the subfunctor consisting of those $(\mathcal{E},\iota) \in \operatorname{Gr}_G(A)$ for which there exist an fppf cover $A\rightarrow A'$ trivialising $\mathcal{E}$ so that, if $\iota \times_{A[u]} A'[u]$ is given by left multiplication by $g \in G(A'[u,E(u)^{-1}])$ , then

$$ E(u) \operatorname{dlog}_u(g) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]. $$

This is well defined because of other choice of trivialisation replaces g by gh for $h \in L^+G$ and Lemma 6.5(i) shows $E(u) \operatorname{dlog}_u(g) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]$ if and only if $E(u) \operatorname{dlog}_u(gh) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]$ . This is a closed subfunctor since $A[u,E(u)^{-1}]/A[u]$ is a projective A-module. When $G = \operatorname{GL}_n$ this coincides with the subfunctor defined in [Reference BartlettBar24, 7.4]. Lemma 6.5 shows that $\operatorname{Gr}^{\nabla}_G$ is also G-stable.

Proof. Since $\operatorname{Gr}^{\nabla}_G$ is a closed subfunctor it suffices to show that $M_\mu \otimes_{\mathcal{O}} E \subset \operatorname{Gr}_G^\nabla$ . For this observe that, under the identification $\operatorname{Gr}_G \otimes_{\mathcal{O}} E \cong ( \operatorname{Gr}_{G,1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} \operatorname{Gr}_{G,e} ) \otimes_{\mathcal{O}} E$ , one has

$$ \operatorname{Gr}_G^\nabla \otimes_{\mathcal{O}} E = ( \operatorname{Gr}_{G,1}^\nabla \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} \operatorname{Gr}_{G,e}^\nabla ) \otimes_{\mathcal{O}} E, $$

where $\operatorname{Gr}_{G,i}^\nabla $ is defined analogously to $\operatorname{Gr}_{G}^\nabla$ with the condition $E(u) \operatorname{dlog}_u(g) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]$ replaced by $(u-\pi_i) \operatorname{dlog}_u(g) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]$ . We are therefore reduced to showing that the closed immersions $G/P_\lambda \rightarrow \operatorname{Gr}_{G,i}$ induced by any $\lambda \in X_*(T)$ factor through $\operatorname{Gr}_{G,i}^\nabla$ , and this follows from Lemma 6.5.

7. Computations in $\operatorname{Gr}_G^\nabla$

In this section we show that the inclusion $M_\mu \subset \operatorname{Gr}_G^\nabla$ induces a reasonable topological description of $M_\mu \otimes \mathbb{F}$ provided $\mu$ is sufficiently small relative to the characteristic of $\mathbb{F}$ . As with the previous section, this extends results from [Reference BartlettBar24, § 7] beyond $G = \operatorname{GL}_n$ .

Proposition 7.1. Suppose that $\lambda \in X_*(T)$ is dominant. If $\operatorname{char}\mathbb{F} >0$ then assume that

$$ \langle \alpha^\vee, \lambda \rangle \leq \operatorname{char}\mathbb{F} + e -1 $$

for every positive root $\alpha^\vee$ . Then

$$ \operatorname{Gr}_{G,\lambda,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^\nabla $$

is smooth and irreducible of dimension

$$ \sum_{\alpha \in R^+} \operatorname{min}\lbrace e, \langle \alpha,\lambda \rangle \rbrace. $$

Proof. As in the previous section we write $\mathfrak{g} = \operatorname{Lie}(G)$ . Let $\mathfrak{t} = \operatorname{Lie}(T)$ and write

$$ \mathfrak{g} = \mathfrak{t} \oplus \bigoplus_{\alpha^\vee\in R^\vee} \mathfrak{g}_{\alpha^\vee} $$

for the root decomposition of $\mathfrak{g}$ . For each $\alpha^\vee \in R^\vee$ we have the associated root homomorphism $x_{\alpha^\vee}: \mathbb{G}_a \rightarrow G$ which induces an identification $dx_{\alpha^\vee}: \mathbb{G}_a \xrightarrow{\sim} \mathfrak{g}_{\alpha^\vee}$ . See, for example, [Reference JantzenJan03, 1.2].

Step 1. We begin by recalling a standard open cover of $G/P_{\lambda}$ . Let U denote the image of the morphism

$$ \prod_{\langle \alpha^\vee, \lambda \rangle >0} \mathbb{A}^1 \rightarrow G, $$

given by $(a_{\alpha^\vee})_{\alpha^\vee} \mapsto \prod_{\alpha^\vee} x_{\alpha^\vee}(a_{\alpha^\vee})$ (the product taken in an arbitrary, but fixed, order). Then U is a closed subgroup of G and the induced morphism $U \rightarrow G/P_\lambda$ is an open immersion. Furthermore, the W-translates of the image of U form an open cover of $G/P_\lambda$ . See [Reference JantzenJan03, 1.10] for more details.

Step 2. Let $U_{\lambda}$ denote the image of the morphism

$$ \prod_{\langle \alpha^\vee,\lambda \rangle >0} \mathbb{A}^{\langle \alpha^\vee,\lambda \rangle} \rightarrow L^+G, $$

given by $(a_{\alpha^\vee,0},a_{\alpha,1},\ldots,a_{\alpha^\vee, \langle \alpha^\vee,\lambda \rangle -1})_{\alpha^\vee} \mapsto \prod_{\alpha^\vee} x_{\alpha^\vee}( \sum_i a_{\alpha^\vee,i} u^i)$ (again the product is taken in an arbitrary, but fixed, order). Then the morphism $U_{\lambda} \rightarrow \operatorname{Gr}_{G,\lambda,\mathbb{F}}$ given by $g \mapsto gu^\lambda$ is an open immersion whose W-translates cover $\operatorname{Gr}_{G,\lambda,\mathbb{F}}$ . To see this note first that $U_\lambda \rightarrow \operatorname{Gr}_{G,\lambda,\mathbb{F}}$ is a monomorphism. Secondly, note that $U_{\lambda} \rightarrow \operatorname{Gr}_{G,\lambda,\mathbb{F}}$ factors through the preimage of U under the morphism $q:\operatorname{Gr}_{G,\lambda,\mathbb{F}} \rightarrow G/P_\lambda$ sending $gu^\lambda$ onto g modulo u. Thirdly, note that $q^{-1}(U)$ is smooth and irreducible of dimension $\sum_{\alpha \in R^+ } \langle \alpha^\vee,\lambda \rangle = \langle 2\rho^\vee,\lambda \rangle$ (because the same is known to be true of $\operatorname{Gr}_{G,\lambda,\mathbb{F}}$ , see for example [Reference ZhuZhu17, 2.1.5]). Therefore $U_\lambda \rightarrow q^{-1}(U)$ , being a monomorphism between integral schemes of the same dimension, is an isomorphism (because monomorphisms are unramified [Sta17, 02GE] and unramified morphisms are etale locally closed immersions [Reference LiuLiu02, 4.11]).

Step 3. We are going to compute the closed subscheme $U_\lambda \times_{\operatorname{Gr}_G} \operatorname{Gr}_G^\nabla$ . By definition $g \in U_\lambda(A)$ is contained in this closed subscheme if and only if

$$ u^e \operatorname{dlog}_u(gu^\lambda) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]. $$

Lemma 6.5 shows this is equivalent to asking that

$$ u^e\operatorname{Ad}(u^{-\lambda}) \operatorname{dlog}_u(g) \in \mathfrak{g} \otimes_{\mathcal{O}} A[u]. $$

It will therefore be necessary to compute $\operatorname{dlog}_u(g)$ and we will do this using the following two observations:

• – If $g = x_{\alpha^\vee}(a)$ for $a \in A[u]$ then $\operatorname{dlog}_u(g) = dx_{\alpha^\vee}(({d}/{du})a)$ . This follows from Example 6.4 and the functoriality of $\operatorname{dlog}_u$ .

• – If $\alpha^\vee +\beta^\vee \neq 0$ then

  (7.2) \begin{equation} \operatorname{Ad}(x_{\alpha^\vee}(a)) dx_{\beta^\vee}(b) = dx_{\beta^\vee}(b) + \sum_{i >0} c_{i1} d_{i\alpha^\vee + \beta^\vee}(a^i b) \end{equation}
  for some $c_{ij} \in \mathbb{Z}$ independent of a and b. This can be seen by passing the formula
  \begin{equation*}\label{eq-commutator} x_{\alpha^\vee}(a) x_{\beta^\vee}(b) x_{\alpha^\vee}(a)^{-1} x_{\beta^\vee}(b)^{-1}= \prod_{i,j>0} x_{i\alpha^\vee + j\beta^\vee}(c_{ij} a^ib_j) \end{equation*}
  found in, for example, [Reference JantzenJan03, 1.2.(5)] to the Lie algebra.

Step 4. Lemma 6.5 shows that $\operatorname{dlog}_u(g x_{\beta^\vee}(b)) = \operatorname{Ad}(x_{\beta^\vee}(-b)) \operatorname{dlog}_u(g) + \operatorname{dlog}_u(x_{\beta^\vee}(b))$ . This, together with the two bullet points from Step 3, allows an inductive computation of $\operatorname{dlog}_u(g)$ for $g = \prod_{\langle \alpha^\vee,\lambda \rangle >0} x_{\alpha^\vee}(a_{\alpha^\vee})$ with $a_{\alpha^\vee} \in A[u]$ . We see that $\operatorname{dlog}_u(g)$ can be expressed as a sum, over $\gamma^\vee$ with $\langle \gamma^\vee,\lambda \rangle >0$ , of terms

$$ dx_{\gamma^\vee}\bigg( \frac{d}{du}a_{\gamma^\vee} + C_{\gamma^\vee} \bigg), $$

where $C_{\gamma^\vee}$ is a $\mathbb{Z}$ -linear combination of products of the $a_{\alpha^\vee}$ and ${d(a_{\alpha^\vee})}/{du}$ for those roots $\alpha^\vee$ with $\langle \alpha^\vee,\lambda \rangle >0$ and $\langle \gamma^\vee - \alpha^\vee,\lambda \rangle >0$ . We can therefore write $C_{\gamma^\vee} = C_{\gamma^\vee,0} + C_{\gamma^\vee,1} u + C_{\gamma^\vee,2} u^2 + \cdots$ with each $C_{\gamma^\vee,i} = C_{\gamma^\vee,i}(a_{\alpha^\vee,j})$ a polynomial in the coefficients of the $a_{\alpha^\vee}$ for $\alpha^\vee$ with $0 < \langle \alpha^\vee,\lambda \rangle < \langle \gamma^\vee,\lambda \rangle$ . These polynomials have $\mathbb{Z}$ -coefficients and depend only on the order in which the product defining g is taken. It follows that $u^e \operatorname{Ad}(u^{-\lambda})\operatorname{dlog}_u(g)$ can likewise be expressed as a sum of the terms

(7.3) \begin{equation}\label{eq-dlogU} u^{e - \langle \gamma^\vee,\lambda \rangle}dx_{\gamma^\vee}\bigg( \frac{d}{du}a_{\gamma^\vee} + C_{\gamma^\vee} \bigg). \end{equation}

The assumption that $\langle \gamma^\vee,\lambda \rangle - e +1\leq \operatorname{char}\mathbb{F}$ means there exist unique polynomials $D_{\gamma^\vee,i} = D_{\gamma^\vee,i}(a_{\alpha^\vee,i})$ in the coefficients of the $a_{\alpha}$ for $\alpha^\vee $ with $0 < \langle \alpha^\vee,\lambda \rangle < \langle \gamma^\vee,\lambda \rangle$ so that if

$$ D_{\gamma^\vee} = D_{\gamma^\vee,1} u + D_{\gamma^\vee,2} u^2 + \cdots + D_{\gamma^\vee,\langle \gamma^\vee,\lambda \rangle-e+1} u^{\langle \gamma^\vee,\lambda \rangle-e+1}, $$

then ${dD_{\gamma^\vee}}/{du} \equiv C_{\gamma^\vee}$ modulo $u^{\langle \gamma^\vee,\lambda \rangle -e+1}$ . Again these polynomials depend only on the order of the product defining g. Thus (7.3) is contained in $\mathfrak{g}_{\gamma^\vee} \otimes_{\mathcal{O}} A[u]$ if and only if

$$ a_{\gamma^\vee} - D_{\gamma^\vee} \in A + u^{\langle \gamma^\vee,\lambda\rangle -e +1}A[u]. $$

It follows that there is an isomorphism

$$ \prod_{\langle \gamma^\vee,\lambda \rangle > 0}\mathbb{A}^{\operatorname{min}\lbrace e,\langle \gamma^\vee,\lambda \rangle\rbrace} \rightarrow U_\lambda \times_{\operatorname{Gr}_G} \operatorname{Gr}^\nabla_G, $$

sending $(a_{\gamma^\vee,i})_{\gamma^\vee}$ onto

$$ \prod_{\gamma^\vee} x_{\gamma^\vee}(a_0 + a_{\gamma^\vee,1} u^{\langle \gamma^\vee,\lambda \rangle -1} + a_{\gamma^\vee,2} u^{\langle \gamma^\vee,\lambda \rangle -2} +\cdots + a_{\gamma^\vee,\operatorname{min}\lbrace e,\langle \gamma^\vee,\lambda \rangle-\rbrace-1} u^{\operatorname{max}\lbrace 1,\langle \gamma^\vee,\lambda \rangle\rbrace -e+1} + D_{\gamma^\vee}). $$

This shows that $U_\lambda \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^{\nabla}$ is smooth of the claimed dimension.

Step 5. It remains to show that $\operatorname{Gr}_{G,\lambda,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^\nabla$ is irreducible. For this recall the action of $\mathbb{G}_m$ on $\operatorname{Gr}_G \otimes_{\mathcal{O}} \mathbb{F}$ via loop rotations: if $t \in A^\times$ and $(\mathcal{E},\iota) \in \operatorname{Gr}_{G,A}$ then

$$ t \cdot (\mathcal{E},\iota) = (x_t^*\mathcal{E},x_t^*\iota), $$

where $x_t$ is the automorphism of $\operatorname{Spec}A[u]$ given by $u \mapsto tu$ . If $\mathcal{E}$ has a trivialisation with respect to which $\iota$ is given by $g(u) \in G(A[u,E(u)^{-1}])$ then $t\cdot(\mathcal{E},\iota)$ corresponds to the trivial G-torsor and $g(tu) \in G(A[u,E(u)])$ . In particular, this demonstrates why this is only an action on $\operatorname{Gr}_G \otimes_{\mathcal{O}} \mathbb{F}$ ; if A is not an $\mathbb{F}$ -algebra then g(ut) need not be in G(A[u,E(u)]).

This action stabilises both $\operatorname{Gr}_{G,\lambda,\mathbb{F}}$ and $\operatorname{Gr}_G^{\nabla}$ . The former stabilisation is clear and the latter follows from the observation, using Remark 6.4 and the chain rule, that $\operatorname{dlog}_u(g(ut)) = t x_t^* \operatorname{dlog}_u(g)$ . The smoothness of $\operatorname{Gr}_{G,\lambda,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^\nabla$ then ensures it is an affine bundle over its $\mathbb{G}_m$ -fixed points, see [Reference MilneMil17, Theorem 13.47]. Since the fixed point locus in $\operatorname{Gr}_{G,\lambda,\mathbb{F}}$ is the G-orbit of $\mathcal{E}_{\lambda,\mathbb{F}}$ , and since this is contained in $\operatorname{Gr}_G^\nabla$ , we conclude that the fixed point locus of $\operatorname{Gr}_{G,\lambda,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^\nabla$ is also this G-orbit. As this orbit is irreducible the same is true for $\operatorname{Gr}_{G,\lambda,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^\nabla$ .

8. Naive cycle identities

Here we use Proposition 7.1 to produce a basic description of the cycles associated with $M_{\mu} \otimes_{\mathcal{O}} \mathbb{F}$ .

Definition 8.1. A d-dimensional cycle on any Noetherian (ind)-scheme X is a $\mathbb{Z}$ -linear combination of integral closed subschemes in X of dimension d. The group of all such cycles is denoted $Z_d(X)$ . If $\mathcal{F}$ is any coherent sheaf on X we write

$$ [\mathcal{F}\,] = \sum_Z m(Z,\mathcal{F}) [Z], $$

where the sum runs over d-dimensional integral closed subschemes Z in X and m(Z,Y) denotes the $\mathcal{O}_{Z,\xi}$ -dimension of $\mathcal{F}_\xi$ for $\xi \in Z$ the generic point. If $i:Y \subset X$ is a closed subscheme then we set $[Y] = [i_*\mathcal{O}_Y]$ . If X is a scheme then [Sta17, 02S9] shows that $Z_d(X)$ can alternatively be defined as the cokernel of the map

$$ K_0(\operatorname{Coh}_{\leq d-1}(X)) \rightarrow K_0(\operatorname{Coh}_{\leq d}(X)), $$

where $\operatorname{Coh}_{\leq d}(X)$ denotes the category of coherent sheaves on X with support of dimension $\leq d$ . Then $[\mathcal{F}\,]$ coincides with the image of the class of $\mathcal{F}$ .

Definition 8.2. For $\lambda \in X_*(T)$ define $\mathcal{C}_\lambda \subset \operatorname{Gr}_G \otimes_{\mathcal{O}} \mathbb{F}$ as the closure of $\operatorname{Gr}_{G,\lambda,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}^\nabla$ . Proposition 7.1 ensures that $\mathcal{C}_\lambda$ is an integral closed subscheme of dimension

$$ \sum_{\text{positive }\alpha^\vee} \operatorname{min}\lbrace e,\langle \alpha^\vee,\lambda \rangle \rbrace $$

provided $\langle \alpha^\vee,\lambda \rangle \leq \operatorname{char}\mathbb{F} +e-1$ for every positive root $\alpha^\vee$ .

Proposition 8.3. Assume that G admits a twisting element $\rho \in X_*(T)$ and suppose that $\mu = (\mu_1,\ldots,\mu_e)$ with $\mu_i \in X_*(T)$ strictly dominant. If $\operatorname{char}\mathbb{F} >0$ assume also that

$$ \sum_i \langle \alpha^\vee,\mu_i \rangle \leq \operatorname{char}\mathbb{F} + e-1 $$

for every positive root $\alpha^\vee$ . Then there exists $m_{\lambda} \in \mathbb{Z}$ so that as $e|R^+|$ -dimensional cycles in $\operatorname{Gr}_{G}\otimes_{\mathcal{O}} \mathbb{F}$

$$ [M_\mu \otimes_{\mathcal{O}}\mathbb{F}] = \sum_{\lambda} m_{\lambda} [\mathcal{C}_{\lambda+e\rho}] $$

with the sum running over dominant $\lambda \leq \mu_1+\cdots +\mu_e - e\rho$ . Furthermore, $m_{\mu_1+\cdots+\mu_e - e\rho}=1$ .

Later on we will give the $m_\lambda$ a representation theoretic interpretation (see Theorem 12.1).

Proof. Propositions 6.7 and 5.9 ensures $M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ factors through $\operatorname{Gr}_{G}^\nabla$ and $Y_{G,\leq \mu}$ . Lemma 5.6 implies $(Y_{G,\leq \mu}\otimes_{\mathcal{O}}\mathbb{F})_{\operatorname{red}} = \bigcup_{\lambda \leq \mu_1+\cdots+\mu_e-e\rho} \operatorname{Gr}_{G,\lambda+e\rho,\mathbb{F}}$ and so

(8.4) \begin{equation}\label{eq-initialcontain} (M_\mu \otimes_{\mathcal{O}} \mathbb{F})_{\operatorname{red}} \subset \bigcup_{\lambda \leq \mu_1+\cdots+\mu_e-e\rho} \operatorname{Gr}_{G,\lambda+e\rho,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G}^\nabla \subset \bigcup_{\lambda \leq \mu_1+\cdots+\mu_e- e\rho} \mathcal{C}_{\lambda+e\rho}. \end{equation}

These unions run over $\lambda$ which are not necessarily dominant. To show that the containment still holds with the union running over dominant $\lambda$ we use the assumption that each $\mu_i$ is strictly dominant. This ensures that $\operatorname{dim}G/P_{\mu_i} = |R|$ and so $\operatorname{dim}M_\mu \otimes_{\mathcal{O}} \mathbb{F} = e|R^+|$ . Thus

$$ \operatorname{dim}\mathcal{C}_{\lambda+e\rho} = \sum_{\text{positive }\alpha^\vee} \operatorname{min}\lbrace e,\langle \alpha^\vee,\lambda +e\rho \rangle \rbrace \leq \operatorname{dim}M_\mu \otimes_{\mathcal{O}} \mathbb{F} $$

with equality if and only if $\langle \alpha^\vee,\lambda+e\rho \rangle \geq e$ for every positive $\alpha^\vee$ . Notice that $\langle \alpha^\vee, \lambda + e\rho \rangle \geq e$ for every positive root $\alpha^\vee$ (equivalently every simple root) if and only if $\lambda$ is dominant, because $\langle \alpha^\vee,\rho \rangle = 1$ whenever $\alpha^\vee$ is simple. Thus, (8.4) can be refined to

$$ (M_\mu \otimes_{\mathcal{O}} \mathbb{F})_{\operatorname{red}} \subset \bigcup \mathcal{C}_{\lambda+e\rho} $$

with the union running over dominant $\lambda \leq \mu_1+\cdots+\mu_e- e\rho$ . In other words,

$$ [M_\mu \otimes_{\mathcal{O}}\mathbb{F}] = \sum_{\lambda'} m_{\lambda} [\mathcal{C}_{\lambda+e\rho}] $$

as desired. To finish the proof we have to $m_{\mu_1+\cdots+\mu_e-e\rho} = 1$ , and for this it suffices to show $\mathcal{C}_{\mu_1+\cdots+\mu_e} \subset M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ and this closed immersion becomes an open immersion after restricting to an open subset of $\mathcal{C}_{\mu_1+\cdots+\mu_e}$ . Recall from Lemma 5.6 that $\operatorname{Gr}_{G,\mu_1+\cdots+\mu_e,\mathbb{F}}$ is open in $Y_{G,\leq \mu} \otimes_{\mathcal{O}} \mathbb{F}$ . Therefore,

$$ U := M_\mu \times_{\operatorname{Gr}_G} \operatorname{Gr}_{G, \mu_1+\cdots+\mu_e,\mathbb{F}} $$

is open in $M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ . It is also non-empty because it is easy to see that $\mathcal{E}_{\mu_1+\cdots+\mu_e,\mathbb{F}} \in M_\mu$ . Therefore, U has dimension equal $\operatorname{dim}M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ . On the other hand U is a closed subscheme of $\operatorname{Gr}_{G,\mu_1+\cdots+\mu_e,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}^\nabla$ . We saw in Proposition 7.1 that $\operatorname{Gr}_{G, \mu_1+\cdots+\mu_e,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}^\nabla$ is smooth and irreducible of the same dimension. Thus, $U = \operatorname{Gr}_{G,\mu_1+\cdots+\mu_e,\mathbb{F}} \times_{\operatorname{Gr}_G} \operatorname{Gr}^\nabla$ . As $\mathcal{C}_{\mu_1+\cdots+\mu_e}$ is the closure of U we conclude that $\mathcal{C}_{\mu_1+\cdots+\mu_e} \subset M_\mu \otimes_{\mathcal{O}} \mathbb{F}$ , and that this inclusion is an isomorphism over an open subset.

9. Irreducibility

Theorem 9.1. Assume that G contains a twisting element $\rho \in X_*(T)$ and suppose that $\lambda \in X_*(T)$ is dominant. If $\operatorname{char}\mathbb{F} >0$ then assume also that

$$ \langle \alpha^\vee,\lambda + e\rho \rangle \leq \operatorname{char}\mathbb{F} + e-1 $$

for every positive root $\alpha^\vee$ . Then, as cycles

$$ [M_{(\lambda+\rho,\rho,\ldots,\rho)} \otimes_{\mathcal{O}} \mathbb{F}] =[\mathcal{C}_{\lambda+e\rho}]. $$

In other words, $M_{(\lambda+\rho,\rho,\ldots,\rho)} \otimes_{\mathcal{O}} \mathbb{F}$ is irreducible and generically reduced.

For this we need the following lemma. Here, we use the notation $Y_{G, \leq \mu_2+\cdots+\mu_e,\mathbb{F}} \subset \operatorname{Gr}_{G} \otimes_{\mathcal{O}} \mathbb{F}$ to denote $Y_{G,\leq \eta} \otimes_{\mathcal{O}} \mathbb{F}$ for any tuple of cocharacters $\eta$ whose sum equals $\mu_2+\cdots+\mu_e$ (this special fibre is independent of $\eta$ by Lemma 5.6).

Lemma 9.2. Suppose $\mu = (\mu_1,\ldots,\mu_e)$ with each $\mu_i \in X^*(T)$ dominant. Then any $\mathbb{F}$ -valued point of $M_{\mu}(\mathbb{F})$ is contained in

$$ g_1 u^{\mu_1} Y_{G, \leq \mu_2+\cdots+\mu_e,\mathbb{F}} $$

for some $g_1 \in G$ .

Proof. Consider the convolution Grassmannian $\operatorname{Gr}^{\operatorname{con}}_{G}$ whose A-points classify pairs $\mathcal{E}_{2},\mathcal{E}_1$ of G-torsors on $\operatorname{Spec}A[u]$ together with isomorphisms $\iota_1:\mathcal{E}_1|_{\operatorname{Spec}A[u,(u-\pi_1)^{-1}]} \xrightarrow{\sim} \mathcal{E}^0|_{\operatorname{Spec}A[u,(u-\pi_1)^{-1}]}$ and

$$ \iota_2: \mathcal{E}_{2}|_{\operatorname{Spec}A\left[u,\prod_{i=2}^e (u-\pi_i)^{-1}\right]}\xrightarrow{\sim} \mathcal{E}_{1}|_{\operatorname{Spec}A\left[u,\prod_{i=2}^e (u-\pi_i)^{-1}\right].} $$

Then $(\mathcal{E}_1,\mathcal{E}_2,\iota_1,\iota_2) \mapsto (\mathcal{E}_2, \iota_1 \circ \iota_2)$ induces a convolution morphism

$$ \operatorname{Gr}^{\operatorname{con}}_G \rightarrow \operatorname{Gr}_G, $$

which is well known to be both proper, and an isomorphism after applying $\otimes_{\mathcal{O}} E$ . In particular, we can define $M^{\operatorname{con}}_\mu \subset \operatorname{Gr}^{\operatorname{con}}_G$ as the closure of $G/P_{\mu_1} \times_E \cdots \times_E G/P_{\mu_e} \subset \operatorname{Gr}^{\operatorname{con}}_G \otimes_{\mathcal{O}} E$ , just as in Definition 4.7. Then convolution restricts to a morphism $M^{\operatorname{con}}_\mu \rightarrow M_\mu$ which is proper and an isomorphism on generic fibres. In particular, this morphism is surjective on $\mathbb{F}$ -valued points.

Now write $\operatorname{Gr}_G^{(e-1)}$ for the affine Grassmannian defined as in Definition 4.1 but for the $e-1$ -tuple $(\pi_2,\ldots,\pi_e)$ . As in Definition 5.2 we can consider the closed subscheme $Y_{G,\leq (\mu_2,\ldots,\mu_e)}$ whose special fibre coincides with $Y_{G,\leq \mu_2+ \cdots+\mu_e,\mathbb{F}}$ . Then we can define a closed subscheme Z of $\operatorname{Gr}^{\operatorname{con}}_G$ by imposing the following conditions.

• – The point of $\operatorname{Gr}_{G,1}$ defined by $(\mathcal{E}_1,\iota_1)$ lies in $G/P_{\mu_1}$ .

• – For any trivialisation $\mathcal{E}_1 \cong \mathcal{E}^0$ over $\operatorname{Spec}A[u]$ the point of $\operatorname{Gr}_{G}^{(e-1)}$ defined by $\mathcal{E}_1$ and the trivialisation

  $$ \mathcal{E}_{2}|_{\operatorname{Spec}A[u,\prod_{i=2}^e (u-\pi_i)^{-1}]}\xrightarrow{\iota_2} \mathcal{E}_{1}|_{\operatorname{Spec}A[u,\prod_{i=2}^e (u-\pi_i)^{-1}]} \cong \mathcal{E}^0|_{\operatorname{Spec}A[u,\prod_{i=2}^e (u-\pi_i)^{-1}]}, $$
  lies in $Y_{G, \leq (\mu_2,\ldots,\mu_e)}$ .

Then $Z \otimes_{\mathcal{O}} E \cong G/P_{\mu_1} \times_E Y_{G,\leq \mu_2} \times_E \cdots \times_E Y_{G, \leq \mu_e}$ and so contains $M^{\operatorname{con}}_\mu \otimes_{\mathcal{O}} E$ . As $Z \subset \operatorname{Gr}^{\operatorname{con}}$ is closed it follows Z contains $M_\mu$ . It therefore suffices to show that any $\mathbb{F}$ -valued point of $\operatorname{Gr}_G$ which is in the image of Z under convolution lies in $g_1 u^{\mu_1} Y_{G, \leq \mu_2+\cdots+\mu_e,\mathbb{F}}$ for some $g_1 \in G$ . But this is clear because if $(\mathcal{E}_1,\mathcal{E}_2,\iota_1,\iota_2) \in Z(\mathbb{F})$ and $\beta: \mathcal{E}_1 \cong \mathcal{E}^0$ is a trivialisation over $\operatorname{Spec}\mathbb{F}[u]$ so that $\iota_1 = g(u-\pi_1)^{\mu_1} \circ \beta$ then convolution sends $(\mathcal{E}_1,\mathcal{E}_2,\iota_1,\iota_2)$ onto

$$ (\mathcal{E}_2,\iota_1\circ \iota_2) = (\mathcal{E}_2, g(u-\pi_1)^{\mu_1} \circ \beta \circ \iota_2) = g(u-\pi_1)^{\mu_1} \cdot (\mathcal{E}_2,\beta \circ \iota_2), $$

and this last point is contained in $g(u-\pi_1)^{\mu_1} Y_{G, \leq (\mu_2,\ldots,\mu_e)} \otimes_{\mathcal{O}} \mathbb{F}$ as desired.

Proof of Theorem 9.1. In view of Proposition 8.3, the theorem will follow if we can show that $\mathcal{C}_{\lambda'+e \rho} \not\subset M_{(\lambda+\rho,\rho,\ldots,\rho)} \otimes_{\mathcal{O}} \mathbb{F}$ for any dominant $\lambda' < \lambda$ . We will do this by exhibiting an $\mathbb{F}$ -valued point $g_0 \in G$ so that

(9.3) \begin{equation}\label{eq-calE} \mathcal{E} = (\mathcal{E}^0, u^{\lambda'+\rho} g_0 u^{(e-1)\rho}) \in \mathcal{C}_{\lambda'+e\rho}, \end{equation}

but cannot be written as in Lemma 9.2. We refer to Example 9.5 for an illustration of some of the key steps in our argument in the case $G = \operatorname{GL}_3$ .

Step 1. We begin by constructing $g_0$ as above. Recall $U \subset B$ denotes the unipotent subgroup and write $H = u \operatorname{dlog}_u(u^{\lambda'+\rho})$ which, by Lemma 6.5, lies in $\operatorname{Lie}(T)$ . We claim there exists $g_0 \in U$ so that

(9.4) \begin{equation}\label{eq-adjointcondition} \operatorname{Ad}(g_0^{-1}) H = H + \sum_{\text{simple }\alpha^\vee} dx_{\alpha^\vee}(A_{\alpha^\vee}) \end{equation}

for some $A_{\alpha^\vee} \in \mathbb{F}^\times$ (recall here that $dx_{\alpha^\vee}: \mathbb{G}_a \rightarrow \mathfrak{g}_{\alpha^\vee}$ is the derivative of the root homomorphism $x_{\alpha^\vee}: \mathbb{G}_a \rightarrow G$ ). To see this, write $g_0^{-1} = \prod_{\beta^\vee >0} x_{\beta\vee}(a_{\beta^\vee})$ with the product running over all positive roots and taken in some fixed order. Then the description of the adjoint action from (7.2) allows us to write

$$ \operatorname{Ad}(g_0^{-1}) H = H + \sum_{\beta^\vee >0} dx_{\beta_\vee}( \beta^\vee(H) a_{\beta^\vee} + P_{\beta^\vee}), $$

where $P_{\beta^\vee}$ is some polynomial in the $a_{\gamma^\vee}$ for roots $\gamma^\vee < \beta^\vee$ . Lemma 6.5 implies $\beta^\vee( H ) = \langle \beta^\vee, \lambda'+ \rho\rangle$ and we claim this is non-zero in $\mathbb{F}$ . Indeed, if $\beta^\vee_{\operatorname{max}}$ is the maximal root above $\beta^\vee$ then

$$ 0 < \beta^\vee( H ) \leq \langle \beta_{\operatorname{max}}^\vee, \lambda'+ \rho\rangle \leq \langle \beta_{\operatorname{max}}^\vee, \lambda+ \rho\rangle, $$

where the first inequality follows from the dominance of $\lambda'$ and the last inequality uses that $\lambda'< \lambda$ and that $\langle \beta^\vee_{\operatorname{max}}, \alpha \rangle \geq 0$ for any simple coroot $\alpha$ (a consequence of the maximality of $\beta^\vee_{\operatorname{max}}$ ). The bound on $\lambda$ in the theorem therefore gives

$$ 0 < \beta^\vee(H) < \operatorname{char}\mathbb{F}, $$

as claimed. Consequently, (9.4) forces $a_{\beta^\vee}$ for non-simple $\beta^\vee$ to be expressed in terms of the $a_{\alpha^\vee}$ for $\alpha^\vee$ simple and implies $a_{\alpha^\vee} \neq 0$ . In particular, there exists $g_0$ satisfying (9.4) as claimed.

Step 2. Next, we show $\mathcal{E} = (\mathcal{E}^0, u^{\lambda'+\rho} g_0 u^{(e-1)\rho}) \in \mathcal{C}_{\lambda'+e\rho} $ for $g_0$ as in Step 1. Since $\lambda'+\rho$ is dominant we have $u^{\lambda'+\rho} g_0 u^{-(\lambda'+\rho)} \in L^+G$ for any $g_0 \in U$ . Therefore, $\mathcal{E} \in \operatorname{Gr}_{G,\lambda'+e\rho,\mathbb{F}}$ and we just need to check when $\mathcal{E} \in \operatorname{Gr}^{\nabla}$ . Since $g_0 \in G$ we have $\operatorname{dlog}_u(g_0) = 0$ . Lemma 6.5 therefore implies $\mathcal{E} \in \operatorname{Gr}^{\nabla}$ if and only if

$$ \operatorname{Ad}(u^{-(e-1)\rho}) ( \operatorname{Ad}(g_0^{-1}) H ) \in u^{-(e-1)}\mathfrak{g}[u]. $$

As $g_0 \in U$ this is equivalent so asking that

$$ \operatorname{Ad}(g_0^{-1}) H \in \operatorname{Lie}(T) \oplus \bigoplus_{\text{simple }\alpha^\vee } \mathfrak{g}_{\alpha^\vee},$$

which is satisfied for $g_0$ as in Step 1.

Step 3. Now suppose $\mathcal{E}$ satisfies the hypothesis of Lemma 9.2. We claim that

$$ \mathcal{E} \in u^{w(\lambda + \rho)} Y_{G,\leq (e-1)\rho, \mathbb{F}} $$

for some $w \in W$ . To see this consider the action of $\mathbb{G}_m \times T$ on $\operatorname{Gr}_{G,\mathbb{F}}$ where T acts by left multiplication and $\mathbb{G}_m$ acts via the loop rotations described in Step 5 of the proof of Proposition 7.1. Since $g_0 \in G$ the 1-parameter subgroup $\mathbb{G}_m \rightarrow \mathbb{G}_m \times T$ given by $t \mapsto (t,t^{-(\lambda'+\rho)})$ induces an action fixing $\mathcal{E}$ . Therefore, this 1-parameter subgroup acts on the non-empty closed subscheme of $G/P_{\lambda+\rho} \subset \operatorname{Gr}_{G,\mathbb{F}}$ consisting of $g_1 \in G/P_{\lambda+\rho}$ with

$$ \mathcal{E} \in g_1 u^{\lambda+\rho} Y_{G, \leq (e-1)\rho,\mathbb{F}}. $$

It follows that this closed subscheme contains $\operatorname{lim}_{t \rightarrow 0} t \cdot g_1$ . Since the loop rotation fixes $G/P_{\lambda+\rho}$ we have $t \cdot g_1 = t^{-(\lambda'+\rho)} g_1$ . The strict dominance of $\lambda'+\rho$ therefore implies $\operatorname{lim}_{t \rightarrow 0} t \cdot g_1$ is a T-fixed point in $G/P_{\lambda+\rho}$ . In other words, we can choose $g_1$ as above representing an element in W, as claimed.

Step 4. Continue to assume $\mathcal{E}$ satisfies the hypothesis of Lemma 9.2. Then Step 3 says

$$ u^{(-w(\lambda+\rho) + \lambda'+\rho)} g_0 u^{(e-1)\rho} \in Y_{G, \leq (e-1)\rho, \mathbb{F}} $$

for some $w \in W$ . In Step 5 we will show any $g_0$ as in Step 1 lies in $B^- w_0 B^-$ for $w_0 \in W$ the longest element and $B^- = P_{\lambda+\rho}$ the Borel opposite B. Here, we explain how this will lead to a contradiction and finish the proof. First, consider the $\mathbb{G}_m$ -action on $u^{(-w(\lambda+\rho) + \lambda'+\rho)} g_0 u^{(e-1)\rho}$ induced by a strictly dominant cocharacter. Since $g_0 \in U$ we have

$$ \operatorname{lim}_{t \rightarrow 0} t \cdot u^{(-w(\lambda+\rho) + \lambda'+\rho)} g_0 u^{(e-1)\rho} = u^{(-w(\lambda+\rho) + \lambda'+\rho)} u^{(e-1)\rho} \in Y_{G,\leq (e-1)\rho,\mathbb{F}}. $$

Therefore $-w(\lambda+\rho) + \lambda'+\rho \leq 0$ . On the other hand, if we consider the $\mathbb{G}_m$ -action induced by a strictly anti-dominant cocharacter then the containment $g_0 \in B^- w_0 B^-$ implies

$$ \operatorname{lim}_{t \rightarrow 0} t \cdot u^{(-w(\lambda+\rho) + \lambda'+\rho)} g_0 u^{(e-1)\rho} = u^{(-w(\lambda+\rho) + \lambda'+\rho)} w_0 u^{(e-1)\rho} \in Y_{G,\leq (e-1)\rho,\mathbb{F}}. $$

Therefore $w_0( -w(\lambda+\rho) + \lambda'+\rho ) \leq 0$ and so $-w(\lambda+\rho) + \lambda'+\rho \geq 0$ . Consequently, $w(\lambda+\rho) = \lambda'+\rho$ which is impossible since $\lambda' < \lambda$ .

Step 5. To conclude choose $s \in W$ so that $g_0^{-1} = b_1 s b_2$ with $b_1\in B^-$ and $b_2 \in U^-$ (the unipotent radical of $B^-$ ). We have to show $s = w_0$ and we will do this by using that, due to our choice of $g_0$ in Step 1, the application of first $\operatorname{Ad}(b_2)$ , then $\operatorname{Ad}(s)$ , and then $\operatorname{Ad}(b_1)$ sends H onto an element of the form

$$ H + \sum_{\text{simple }\alpha^\vee} dx_{\alpha^\vee}(A_{\alpha^\vee}), $$

with $A_{\alpha^\vee} \in \mathbb{F}^\times$ . First, we can write

$$ \operatorname{Ad}(sb_2)H = s(H) + \sum_{\beta^\vee<0 } dx_{s(\beta^\vee)}(b_{\beta^\vee}) $$

for some $b_{\beta^\vee} \in \mathbb{F}$ . Now, $\operatorname{Ad}(b_1)$ maps $dx_{\alpha^\vee}(1)$ onto a linear combination of $dx_{\beta^\vee}$ values with $\beta^\vee \leq \alpha^\vee$ and the coefficient of $\alpha^\vee$ non-zero. Consequently, if $\gamma^\vee > 0 $ is a root maximal for the property that $b_{s^{-1}(\gamma^\vee)} \neq 0$ then the projection of

$$ \operatorname{Ad}(b_1)\bigg( H + \sum_{\beta^\vee<0 } dx_{s(\beta^\vee)}(b_{\beta^\vee})\bigg) $$

to the $\gamma^\vee$ -root subspace will be a non-zero multiple of $d_{\gamma^\vee}(b_{s^{-1}(\gamma^\vee)})$ . This forces $\gamma^\vee$ to be simple. On the other hand, if $\alpha^\vee$ is simple then $b_{s^{-1}(\alpha^\vee)} \neq 0$ because otherwise the projection of $\operatorname{Ad}(b_1)( H + \sum_{\beta^\vee<0 } dx_{s(\beta^\vee)}(b_{\beta^\vee}))$ onto the $\alpha^\vee$ -root subspace will be zero. Therefore, $s^{-1}(\alpha^\vee) <0$ for each simple $\alpha^\vee >0$ . This implies $s = w_0$ , which finishes the proof.

Example 9.5. Suppose $G = \operatorname{GL}_3$ and write $H = \operatorname{diag}(a,b,c)$ for integers $a> b > c$ . Then $g_0 = \left( \begin{smallmatrix} 1 &\ x &\ z \\ 0 &\ 1 &\ y \\ 0 &\ 0 &\ 1 \end{smallmatrix} \right) \in U$ satisfies the conditions imposed in Step 1 precisely when $z = -xy {(c-b)}/{(a-c)}$ and $x,y \neq 0$ . In this case it is easy to see directly that $g_0 \in B^- w_0 B^-$ as claimed in Step 4. However, we can also use run the arguments from Step 5 in this example to illustrate the idea. For example, if $g_0^{-1} = b_1 s b_2$ for $b_i \in B^-$ and $s = \left( \begin{smallmatrix} 0 &\ 1 &\ 0 \\ 0 &\ 0 &\ 1 \\ 1 &\ 0 &\ 0 \end{smallmatrix} \right)$ then $\operatorname{Ad}(sb_2)$ would send H onto a matrix of the form

$$\begin{pmatrix} * & *_1 & *_2 \\ * & * & 0 \\ * & * & * \end{pmatrix}\!, $$

with the $*$ symbols possibly zero. We would then need $\operatorname{Ad}(b_1)$ to send this matrix onto something with non-zero entries directly above the diagonal and top right entry zero. Considering how $\operatorname{Ad}(b_1)$ acts shows that $*_2 =0$ , otherwise the top right entry could never be killed. On the other hand, if $*_2 =0$ then it is impossible for an application of $\operatorname{Ad}(b_1)$ to create a non-zero entry in the (2,3)th position.

10. Equivariant sheaves and their cycles

Our goal is now to give the coefficients appearing in Proposition 8.3 a representation theoretic meaning. To do this we will relate these cycle identities to relations between global sections of line bundles on the $M_\mu$ .

Suppose X is a finite type $\mathbb{F}$ -scheme equipped with an action of the torus T. Then we write $K_0^T(X)$ for the Grothendieck group of the category of T-equivariant coherent sheaves on X. If X is proper over $\mathbb{F}$ then the Euler characteristic $[\mathcal{F}\,] \mapsto \sum_{i \geq 0} (-1)^i [H^i(X,\mathcal{F})]$ defines a homomorphism

$$ \chi: K_0^T(X) \rightarrow R(T),$$

where $R(T) = \mathbb{Z}[X^*(T)]$ denotes the Grothendieck group of algebraic T-representations (in which multiplication is given by the tensor product). For $\alpha^\vee \in X^*(T)$ we write $e(\alpha^\vee)$ for its class in R(T) and for $V \in R(T)$ we write $V_{\alpha^\vee} \in \mathbb{Z}$ for the multiplicity of $e(\alpha^\vee)$ in V.

• – Define $K_0^T(X)_{\leq d}$ as the Grothendieck group of the category of T-equivariant coherent sheaves on X with support of dimension $\leq d$ .

• – Let $V = (V_n)_{n \geq 0}$ be a sequence of elements in R(T). We say V is polynomial of degree $\leq d$ if there exists a polynomial $P(x) \in \mathbb{Q}[x]$ of degree $\leq d$ so that

  $$ \sum_{\alpha^\vee \in X^*(T)} |V_{n,\alpha^\vee}| \leq P(n) $$
  for all $n\geq 0$ (here $|\cdot|$ denotes the usual absolute value).

Notice that if each $V_n$ is effective (i.e. is the class of a T-representation $V_n$ in R(T)) then V is polynomial if and only if the dimensions of $V_n$ are bounded by the value at n of a polynomial. However, the above definition also allows us to extend this notion to elements which are not necessarily effective.

Lemma 10.4. Suppose that X is a proper $\mathbb{F}$ -scheme of finite type equipped with a T-equivariant ample line bundle $\mathcal{L}$ and $\mathcal{F} \in K_0^T(X)$ is contained in the image of $\operatorname{im}K_0^T(X)_{\leq d}$ . Then

$$ \chi(\mathcal{F} \otimes [\mathcal{L}^{\otimes n}] ) \in R(T) $$

is polynomial of degree $ \leq d$ .

Proof. Using Lemma 10.3 we can assume that $\mathcal{F}$ is the class of a T-equivariant sheaf $\mathcal{G}$ on X with support of dimension $\leq d$ . Since $\mathcal{L}$ is ample $\chi([\mathcal{G} \otimes \mathcal{L}^{\otimes n}])$ equals the class of $H^0(X,\mathcal{G} \otimes \mathcal{L}^{\otimes n})$ in R(T) for sufficiently large n. Since the dimension of $H^0(X,\mathcal{G} \otimes \mathcal{L}^{\otimes n})$ is for sufficiently large n the value at n of a polynomial P(x) of degree equal the support of $\mathcal{G}$ it follows that for all $n \geq 0$

$$ \sum_{\alpha^\vee} |\chi(\mathcal{G} \otimes \mathcal{L}^{\otimes n})_{\alpha^\vee}| = \sum_{\alpha^\vee} \chi(\mathcal{G} \otimes \mathcal{L}^{\otimes n})_{\alpha^\vee} \leq P(n) + C $$

for a constant $C>>0$ .

Proposition 10.5. Let $\mathcal{F}$ be a T-equivariant coherent sheaf on X with support of dimension $\leq d$ and let

$$ [\mathcal{F}\,] = \sum_Z n_Z [\mathcal{O}_Z] \in Z_d(X) $$

be the associated d-dimensional cycle. Assume that X is equipped with an ample T-equivariant line bundle $\mathcal{L}$ and that $n_Z >0$ implies Z is T-stable. Then there are $q_Z \in \mathbb{Z}_{\geq 0}$ and $\theta_{Z,i}^\vee \in X^*(T)$ so that

$$ \chi(\mathcal{F} \otimes \mathcal{L}^{\otimes n}) - \sum_Z \bigg( \sum_{i=1}^{n_Z} e(\theta_{Z,i}^\vee) \chi(\mathcal{L}^{\otimes k - q_Z}|_{Z}) \bigg) \in R(T) $$

is polynomial of degree $< d$ .

Proof. We induct on the number of $n_Z >0$ . If this is zero then the class of $\mathcal{F}$ has support of dimension $<d$ and so its class in $K_0^T(X)$ is contained in $\operatorname{im}K_0^T(X)_{\leq d-1}$ . Lemma 10.4 therefore implies $\chi(\mathcal{F} \otimes \mathcal{L}^{\otimes n})$ is polynomial of degree $<d$ , and the proposition holds. Otherwise, write $\mathcal{I}_Z$ for the ideal sheaves corresponding to those Z with $n_Z >0$ . By assumption each such Z is T-stable and so $\mathcal{I}_Z^N$ is a T-equivariant coherent sheaf for any $N\geq 1$ . For N sufficiently large the support of $\mathcal{I}_Z^N\mathcal{F}$ does not contain Z (see [Sta17, 0Y19]) and so the inductive hypothesis holds for $\mathcal{I}_Z^N\mathcal{F}$ . By applying Lemma 10.4 to the identity

$$ [\mathcal{F}\,] = [\mathcal{I}_Z^N\mathcal{F}] + [\mathcal{F}/\mathcal{I}_Z^N\mathcal{F}] $$

in $K_0^T(X)$ we see that the proposition will hold if it holds for $\mathcal{F}/\mathcal{I}_Z^N\mathcal{F}$ .

Since $\mathcal{F}/I_Z^N\mathcal{F}$ has support contained in Z we are reduced to proving the proposition when the support of $\mathcal{F}$ is a single irreducible component. Thus we can assume $Z= X$ and write $[\mathcal{F}\,] = n[\mathcal{O}_X]$ in $Z_d(X)$ . Since $\mathcal{L}$ is ample there is an integer $q \in \mathbb{Z}_{\geq 0}$ so that $\mathcal{F} \otimes \mathcal{L}^{\otimes q}$ is generated by global sections. This gives a T-equivariant surjection

$$V \otimes \mathcal{L}^{\otimes -q} \rightarrow \mathcal{F,} $$

with $V = H^0(X,\mathcal{F} \otimes \mathcal{L}^{\otimes q})$ . Since T is abelian we can choose a T-equivariant filtration

$$ \cdots\subset V_{j+1} \subset V_j \subset V_{j-1} \subset \cdots V_0 =V, $$

with each graded piece of dimension one.

Claim. There exists $j_1,\ldots,j_n \in \mathbb{Z}_{\geq 0}$ so that

$$ [\mathcal{F}\,]- \sum_{i=1}^n [V_{j_i}/V_{j_i+1} \otimes \mathcal{L}^{-q}] \in \operatorname{im}K_0^T(X)_{\leq d-1}. $$

Proof of Claim. Set $\mathcal{F}_j$ equal the image in $\mathcal{F}$ of $V_j \otimes \mathcal{L}^{-q}$ . Then $[\mathcal{F}\,] = \sum_{j \in \mathbb{Z}_{\geq 0}} [\mathcal{F}_j/\mathcal{F}_{j+1}]$ in $K_0^T(X)$ . For each j we also have a T-equivariant exact sequence

$$ 0 \rightarrow \mathcal{G}_j \rightarrow V_j/V_{j+1} \otimes \mathcal{L}^{-q} \rightarrow \mathcal{F}_j/\mathcal{F}_{j+1} \rightarrow 0 $$

of coherent sheaves. Since $V_j/V_{j+1} \otimes \mathcal{L}^{-q}$ is locally free of rank one it follows that exactly one of $\mathcal{G}_j$ and $\mathcal{F}_j/\mathcal{F}_{j+1}$ has support of dimension $<d$ and exactly one has support equal to X. Thus

$$ \begin{aligned} &[\mathcal{F}\,] - \sum_{\operatorname{supp}\mathcal{F}_j/\mathcal{F}_{j+1} =X} [V_j/V_{j+1}\otimes \mathcal{L}^{\otimes -q}] \\ &\quad = - \sum_{\operatorname{supp}\mathcal{F}_j/\mathcal{F}_{j+1} =X} [\mathcal{G}_j] + \sum_{\operatorname{supp}\mathcal{F}_j/\mathcal{F}_{j+1} \neq X} [\mathcal{F}_j/\mathcal{F}_{j+1}] \in \operatorname{im}K_0^T(X)_{\leq d-1}. \end{aligned} $$

To finish the proof of the claim we just need to check that $\operatorname{supp}\mathcal{F}_j/\mathcal{F}_{j+1} =X$ exactly n times. This follows because in $Z_d(X)$ we have $ [V_j/V_{j+1}\otimes \mathcal{L}^{\otimes -q}] = [X]$ and so $[\mathcal{F}\,] = \sum_{\operatorname{supp}\mathcal{F}_j/\mathcal{F}_{j+1} =X} [V_j/V_{j+1}\otimes \mathcal{L}^{\otimes -q}] = \sum_{\operatorname{supp}\mathcal{F}_j/\mathcal{F}_{j+1} = X} [X] = n[X]$ .

Applying Lemma 10.4 to the identity in the claim gives the proposition because if $\theta_i^\vee \in X^*(T)$ is the character through which T acts on $V_{j_i}/V_{j_i+1}$ then $\chi(V_j/V_{j+1}\otimes \mathcal{L}^{\otimes k-q}) = e(\theta_i^\vee)\chi(\mathcal{L}^{\otimes (k-q)})$ .

11. Determinant line bundles

In order to apply Proposition 10.5 to the identity of cycles established in Proposition 8.3 we need to choose an equivariant line bundle on $\operatorname{Gr}_G$ . To do this we consider the morphism

$$\operatorname{Ad}: \operatorname{Gr}_G \rightarrow\operatorname{Gr}_{\operatorname{GL}(\mathfrak{g})},$$

induced by the adjoint representation of G. Then $\operatorname{Gr}_{\operatorname{GL}(\mathfrak{g})}$ is equipped with the ‘determinantal’ line bundle $\mathcal{L}_{\operatorname{det}}$ , defined by the property that its pull-back to $\operatorname{Spec}A$ along a morphism corresponding to $(\mathcal{E},\iota) \in \operatorname{Gr}_{\operatorname{GL}(\mathfrak{g})}(A)$ is given by the A-module

$$ \operatorname{det}_A(u^{-N} \mathfrak{g} \otimes_{\mathcal{O}} A[u]/ \iota(\mathcal{E})) \otimes_A \operatorname{det}_A(u^{-N}\mathfrak{g} \otimes_{\mathcal{O}} A[u]/ \mathfrak{g} \otimes_{\mathcal{O}} A[u])^{-1} $$

for N sufficiently large that $\iota(\mathcal{E}) \subset u^{-N} \mathfrak{g} \otimes_{\mathcal{O}} A[u]$ . Note this is independent of the choice of N. Then $\mathcal{L}_{\operatorname{det}}$ is $\operatorname{GL}(\mathfrak{g})$ -equivariant and is ample in the sense that its restriction to any closed subscheme in $\operatorname{Gr}_{\operatorname{GL}(\mathfrak{g})}$ is ample. Therefore

$$ \mathcal{L}_{\operatorname{ad}} := \operatorname{Ad}^* \mathcal{L}_{\operatorname{det}} $$

is G-equivariant and also ample.

Lemma 11.1. For $\lambda \in X_*(T)$ the group T acts on the fibre of $\mathcal{L}_{\operatorname{ad}}$ over $\mathcal{E}_{\lambda,i}$ via the image of $\lambda$ under the homomorphism

$$p: X_*(T) \rightarrow X^*(T),$$

given by $\lambda \mapsto \sum_{\alpha^\vee \in R^\vee} \langle \alpha^\vee,\lambda \rangle \alpha^\vee$ .

Proof. This fibre is the rank one $\mathcal{O}$ -module

(11.2) \begin{equation}\label{eq-fibre} \underbrace{\operatorname{det}_{\mathcal{O}}(u^{-N} \mathfrak{g}[u] / \operatorname{Ad}(u-\pi_i)^\lambda \mathfrak{g}[u] )}_{\Lambda_1} \otimes \underbrace{\operatorname{det}_{\mathcal{O}}(u^{-N} \mathfrak{g}[u] / \mathfrak{g}[u] )^{-1}}_{\Lambda_2}\end{equation}

for sufficiently large N. As an $\mathcal{O}$ -module we have

$$u^{-N} \mathfrak{g}[u] / \operatorname{Ad}(u-\pi_i)^\lambda \mathfrak{g}[u] \cong \bigoplus_{\alpha^\vee \in R^\vee} \bigoplus_{n=-N}^{\langle \alpha^\vee,\lambda \rangle} (u-\pi_i)^n \mathfrak{g}_{\alpha^\vee},$$

and $t \in T(\mathcal{O})$ acts on $(u-\pi_i)^n \mathfrak{g}_{\alpha^\vee}$ by $\alpha^\vee(t)$ . Therefore, t acts on $\Lambda_1$ by $\prod_{\alpha^\vee \in R} \alpha^\vee(t)^{\langle \alpha^\vee,\lambda \rangle + N}$ . Similarly t acts on $\Lambda_2$ by $\prod_{\alpha^\vee \in R} \alpha^\vee(t)^{N}$ . We conclude that t acts on (11.2) by

$$\prod_{\alpha^\vee \in R} \alpha^\vee(t)^{\langle \alpha^\vee,\lambda \rangle} = p(\lambda)(t)$$

as claimed.

Proposition 11.3. If $\mu = (\mu_1,\ldots,\mu_e)$ with $\mu_i \in X_*(T)$ dominant then, after interpreting $H^0(M_\mu \otimes_{\mathcal{O}} \mathbb{F},\mathcal{L}_{\operatorname{ad}}^{\otimes n})$ and $\operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i))$ as T-representations by restricting the G-action, one has

$$ [H^0(M_\mu \otimes_{\mathcal{O}} \mathbb{F},\mathcal{L}_{\operatorname{ad}}^{\otimes n})] = \bigg[\bigotimes_{i=1}^e \operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i))\bigg] \in R(T) $$

for sufficiently large $n\geq 0$ .

Proof. Since $\mathcal{L}_{\operatorname{ad}}$ is an ample line bundle on the flat $\mathcal{O}$ -scheme $M_\mu$ it follows that

$$ H^0(M_\mu \otimes_{\mathcal{O}} \mathbb{F},\mathcal{L}_{\operatorname{ad}}^{\otimes n}) = H^0(M_\mu,\mathcal{L}_{\operatorname{ad}}^{\otimes n}) \otimes_{\mathcal{O}} \mathbb{F} $$

for sufficiently large n. Therefore $[H^0(M_\mu \otimes_{\mathcal{O}} \mathbb{F},\mathcal{L}_{\operatorname{ad}}^{\otimes n}) ]$ is equal to the image of $[H^0(M_\mu \otimes_{\mathcal{O}} E,\mathcal{L}_{\operatorname{ad}}^{\otimes n}) ]$ under the specialisation map from [Reference JantzenJan03, 10.9]. Since this map sends the class of $\bigotimes_{i=1}^e \operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i))$ viewed as a representation on an E-vector space onto $\bigotimes_{i=1}^e \operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i))$ viewed as a representation on an $\mathbb{F}$ -vector space the proposition will follow if we can show

$$ [H^0(M_\mu \otimes_{\mathcal{O}} E,\mathcal{L}_{\operatorname{ad}}^{\otimes n})] = \bigg[\bigotimes_{i=1}^e \operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i))\bigg] \in R(T). $$

Under the isomorphism $\operatorname{Gr}_G \otimes_{\mathcal{O}} E \cong ( \operatorname{Gr}_{G,1} \times_{\mathcal{O}} \cdots \times_{\mathcal{O}} \operatorname{Gr}_{G,e} ) \otimes_{\mathcal{O}} E$ we have $\mathcal{L}_{\operatorname{ad}} = \bigotimes_{i=1}^e p_i^* \mathcal{L}_{\operatorname{ad},i}$ where $\mathcal{L}_{\operatorname{ad},i}$ is the restriction of $\mathcal{L}_{\operatorname{ad}}$ to $\operatorname{Gr}_{G,i}$ and $p_i$ is the ith projection. Therefore

$$ \mathcal{L}_{\operatorname{ad}}|_{M_\mu \otimes_{\mathcal{O}} E} = \bigotimes_{i=1}^e p_i^*(\mathcal{L}_{\operatorname{ad,i}}|_{G/P_{\mu_i} \times_{\mathcal{O}} E}),$$

and so the Kunneth formula [Sta17, 0BED] identifies

$$ H^0(M_\mu \otimes_{\mathcal{O}} E,\mathcal{L}_{\operatorname{ad}}^{\otimes n}) = \bigotimes_{i=1}^e H^0(G/P_{\mu_i} \otimes_{\mathcal{O}} E, \mathcal{L}_{\operatorname{ad},i}^{\otimes n}) $$

as G-representations. To finish the proof we just have to show

(11.4) \begin{equation}\label{eq-desiredisom} H^0(G/P_{\mu_i}, \mathcal{L}_{\operatorname{ad},i}^{\otimes n}) \cong \operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i)). \end{equation}

For this recall (see, for example, [Reference JantzenJan03, 5.12]) that the global sections of any G-equivariant line bundle on $G/P_{\mu_i}$ are G-equivariantly isomorphic to $\operatorname{Ind}_{P_{\mu_i}}^G(\eta)$ where $\eta \in X^*(T)$ is the character through which T acts on the fibre over $1 \in G/P_{\mu_i}$ . Since 1 is mapped onto $\mathcal{E}_{\mu_i,i}$ under the closed immersion $G/P_{\mu_i} \rightarrow \operatorname{Gr}_{G,i}$ we deduce (11.4) from Lemma 11.1.

12. Main theorem

For any dominant $\lambda^\vee \in X^*(T)$ write

$$W(\lambda^\vee) = \operatorname{Ind}_{B^-}^{G}(\lambda^\vee),$$

for $B^-$ the Borel opposite to B. Likewise, for $\lambda \in X_*(T)$ we make sense of $W(\lambda)$ , now as a representation of $\widehat{G}$ . The following is an alternative formulation of Theorem 1.1.

Theorem 12.1. Assume that G admits a twisting element $\rho \in X_*(T)$ and let $\mu = (\mu_1,\ldots,\mu_e)$ with each $\mu_i \in X_*(T)$ strictly dominant. If $\operatorname{char}\mathbb{F}>0$ assume also that

$$ \sum_{i=1}^e \langle \alpha^\vee,\mu_i \rangle \leq \operatorname{char}\mathbb{F} + e-1 $$

for all positive roots $\alpha^\vee$ . Then

$$ [M_\mu \otimes_{\mathcal{O}} \mathbb{F}] = \sum m_\lambda [M_{(\lambda+\rho,\ldots,\rho)}] $$

as $e|R^+|$ -dimensional cycles for $m_\lambda \in \mathbb{Z}_{\geq 0}$ determined by the identity

$$\bigg[ \bigotimes_{i=1}^e W(\mu_i-\rho)\bigg] = \sum_{\lambda}m_\lambda [ W(\lambda)] $$

in the Grothendieck group of $\widehat{G}$ -representations.

Proof. We give the proof here using two representation-theoretic propositions from the next section (Propositions 13.2 and 13.4). Proposition 8.3 and Theorem 9.1 imply

$$[M_\mu \otimes_{\mathcal{O}}\mathbb{F}] = \sum n_{\lambda} [M_{\widetilde{\lambda}}\otimes_{\mathcal{O}} \mathbb{F}],$$

where the sum suns over dominant $\lambda \leq \mu_1+\cdots+\mu_e - e\rho$ , $\widetilde{\lambda} = (\lambda+\rho,\rho,\ldots,\rho)$ , and $n_{\lambda} \in \mathbb{Z}_{\geq 0}$ . We have to show that $n_\lambda =m_\lambda$ . Applying Proposition 10.5 to this identity with $\mathcal{L}$ equal to $\mathcal{L}_{\operatorname{ad}}$ gives $\theta_{\lambda,i}^\vee \in X^*(T)$ so that

$$\chi(\mathcal{L}_{\operatorname{ad}}^{\otimes n}|_{M_\mu \otimes_{\mathcal{O}} \mathbb{F}}) - \sum_{\lambda} \sum_{i=1}^{n_\lambda} e(\theta_{\lambda,i}^\vee) \chi(\mathcal{L}_{\operatorname{ad}}^{\otimes n}|_{M_{\widetilde{\lambda}} \otimes_{\mathcal{O}} \mathbb{F}})$$

is polynomial of degree $< e|R_+^\vee|$ . Since each $\mu_i$ is strictly dominant each $P_{\mu_i}$ equals the opposite Borel $B^-$ and so

$$W(p(n\mu_i)) = \operatorname{Ind}_{P_{\mu_i}}^G(p(n\mu_i)).$$

Therefore Proposition 11.3 gives that

\begin{equation*}\label{eq-polynomial<di} \prod_{i=1}^e W(p(n\mu_i)) - \sum_{\lambda} \sum_{i=1}^{n_\lambda} e(\theta_{\lambda,i}^\vee) W(p(n(\lambda+\rho))) W(p(n\rho))^{e-1}\end{equation*}

is polynomial of degree $< e|R_+^\vee|$ . In the next section we prove (see Proposition 13.2) that

$$ \prod_{i=1}^e W(p(n\mu_i)) - \sum_{\lambda} m_\lambda W(p(n(\lambda+\rho))) W(p(n\rho))^{e-1}$$

is polynomial of degree $< e|R_+^\vee|$ , and so considering the difference gives that

$$\sum_\lambda ( m_\lambda - \sum_{i=1}^{n_\lambda} e(\theta_{\lambda,i}^\vee)) W(p(n(\lambda+\rho))) W(p(n\rho))^{e-1}$$

is also polynomial of degree $< e|R_+^\vee|$ . In the next section we also prove (see Proposition 13.4) that if $X_{\lambda} \in R(T)$ are such that

$$\sum_{\lambda} X_{\lambda} W(p(n(\lambda+\rho))) W(p(n\rho))^{e-1}$$

is polynomial of degree $<e|R_+^\vee|$ then $X_\lambda =0$ for each $\lambda$ . Therefore

$$m_\lambda - \sum_{i=1}^{n_\lambda} e(\theta_{\lambda,i}^\vee) =0$$

for each $\lambda$ . This implies that each $\theta^\vee_{\lambda,i} =1$ and that $n_\lambda = m_\lambda$ for each $\lambda$ . The latter assertion, in particular, proves the theorem.

13. Some representation theory

It remains to prove Propositions 13.2 and 13.4 which were used in the proof of Theorem 9.1. For this set $\rho^\vee = ({1}/{2})\sum_{\alpha^\vee \in R_+^\vee} \alpha^\vee$ . Then the Weyl character formula asserts that for any dominant $\lambda^\vee \in X^*(T)$

$$[W(\lambda^\vee)] = \frac{A(\lambda^\vee+\rho^\vee)}{A(\rho^\vee)},$$

where $A(\lambda^\vee) := \sum_{w\ \in W} (-1)^{l(w)} e(w(\lambda^\vee))$ and this identity is occurring inside the ring $\mathbb{Z}[({1}/{2})X^*(T)]$ . See, for example, [Reference JantzenJan03, 5.10].

Lemma 13.1. If $\mu \in X^*(T)$ is strictly dominant then

$$ W(n\mu^\vee) - e(\rho^\vee)\frac{A(n\mu^\vee)}{A(\rho^\vee)} \in R(T) $$

is a sequence of effective elements (i.e. $\mathbb{Z}_{\geq 0}$ -linear combinations of the $e(\alpha^\vee)$ ) which is polynomial in n of degree $<|R^+|$ .

Proof. We first reduce to the case where $X^*(T)$ contains a twisting element $\rho^\vee_0$ , in the sense of Definition 2.1. The construction from [Reference Buzzard and GeeBG14, § 5.3] produces a central extension $1 \rightarrow \mathbb{G}_m \rightarrow \widetilde{G} \rightarrow G \rightarrow 1$ such that if $\widetilde{T} \subset \widetilde{G}$ is the preimage of T then $X^*(\widetilde{T})$ contains such a twisting element. Being a central extension, the Weyl group of $\widetilde{G}$ relative to $\widetilde{T}$ equals W. Therefore, the inclusion

$$ X^*(T) \rightarrow X^*(\widetilde{T}) $$

maps $A(\lambda^\vee)$ onto $A(\widetilde{\lambda}^\vee)$ for $\widetilde{\lambda}^\vee$ the character of $\widetilde{T}$ induced by $\lambda^\vee$ . As a result the lemma holds for G if it holds for $\widetilde{G}$

We can therefore assume there exists a twisting element $\rho_0^\vee \in X^*(T)$ . Then $\rho_0^\vee - \rho^\vee$ is W-invariant and so the Weyl character formula implies $[W(\mu^\vee)] = {(A(\mu^\vee+\rho_0^\vee))}/{A(\rho_0^\vee)}$ . Now, for any $\lambda^\vee \in X^*(T)$ write $\mathcal{L}(\lambda^\vee)$ for the G-equivariant line bundle on $G/B^-$ on which the action of T on the fibre over the identity is given by $\lambda^\vee$ . Then $\mathcal{L}(\rho^\vee_0)$ admits a unique global section on which T acts by $\rho^{\vee}_0$ , and this induces a T-equivariant injection

$$ \mathcal{O}_{G/B^-} \otimes \rho^\vee_0 \hookrightarrow \mathcal{L}_{\rho^\vee_0}. $$

Tensoring with $\mathcal{L}_{\mu^\vee - \rho^\vee_0}$ produces a T-equivariant injection $\mathcal{L}_{\mu^\vee - \rho^\vee_0} \otimes \rho^\vee_0 \hookrightarrow \mathcal{L}_{\mu^\vee}$ and taking global sections yields a T-equivariant injection of $W(\mu^\vee-\rho^\vee_0) \otimes\rho^\vee_0$ into $W(\mu^\vee)$ . In particular,

$$ [W(n\mu^\vee) ]- e(\rho^\vee_0)[W(n\mu^\vee - \rho^\vee_0)] = [W(n\mu^\vee)] - e(\rho^\vee_0) \frac{A(n\mu^\vee)}{A(\rho_0^\vee)} = [W(n\mu^\vee)] - e(\rho^\vee )\frac{A(n\mu^\vee)}{A(\rho^\vee)} $$

is an effective element of R(T) for each $n \geq 0$ . Since it is effective we can show the sequence of elements is polynomial of degree $<|R^+|$ by showing that the difference between the dimensions of $W(n\mu^\vee)$ and $W(n\mu^\vee - \rho^\vee_0)$ is a polynomial in n of degree $<|R^+|$ . But this follow from the Weyl dimension formula $\operatorname{dim} W(\lambda^\vee) = \prod_{\alpha \in R_+} ({\langle \lambda^\vee+\rho^\vee_0,\alpha\rangle}/{\langle \rho^\vee_0,\alpha \rangle})$ since it shows both $W(n\mu^\vee)$ and $W(n\mu^\vee - \rho^\vee_0)$ have dimension the value at n of a degree $|R^+|$ polynomial with leading term $n^{|R^+|}\prod_{\alpha \in R_+} ({\langle \mu^\vee,\alpha \rangle }/{\langle \rho^\vee_0,\alpha \rangle})$ .

Proposition 13.2. Suppose that $X_*(T)$ contains a twisting element $\rho$ and consider strictly dominant $\mu_1,\ldots,\mu_e \in X_*(T)$ . If

$$ \bigg[ \bigotimes_{i=1}^e W(\mu_i-\rho)\bigg] = \sum_{\lambda} m_\lambda[ W(\lambda)] $$

in the Grothendieck group of $\widehat{G}$ -representations then

$$ \prod_{i=1}^e W(p(n\mu_i)) - \sum_{\lambda} m_\lambda W(p(n(\lambda+\rho))) W(p(n\rho))^{e-1} $$

is polynomial of degree $<e|R_*^\vee|$ .

Proof. The Weyl character formula (applied to $\widehat{G}$ ) yields the identity

$$ \prod_{i=1}^e \frac{A(\mu_i)}{A(\rho)} = \sum_{\lambda} m_\lambda \frac{A(\lambda+\rho)}{A(\rho)} $$

in $R(\widehat{T})$ . Multiplying by $A(\rho)^{e}$ gives

$$ \prod_{i=1}^e A(\mu_i) = \sum_{\lambda} m_\lambda A(\lambda+\rho) A(\rho)^{e-1}. $$

The endomorphism of $R(\widehat{T}) = \mathbb{Z}[X_*(T)]$ induced by multiplication by n on $X^*(T)$ is W-equivariant and so commutes with the formation of $A(\lambda)$ . Applying this endomorphism to the previous identity gives

$$ \prod_{i=1}^e A(n\mu_i) = \sum_{\lambda} m_\lambda A(n(\lambda+\rho)) A(n\rho)^{e-1}. $$

The homomorphism $p: X_*(T) \rightarrow X^*(T)$ induces a homomorphism $R(T^\vee) \rightarrow R(T)$ which is again W-equivariant and so also commutes with the formation of $A(\lambda)$ . Therefore, applying this homomorphism and multiplying by $({e(\rho^\vee)}/{A(\rho^\vee)})^e$ gives

(13.3) \begin{equation}\label{eq-finalmult} \prod_{i=1}^e\frac{ e(\rho^\vee)A(np(\mu_i)) }{A(\rho^\vee)}= \sum_{\lambda} m_\lambda \frac{e(\rho^\vee)A(np(\lambda+\rho))}{A(\rho^\vee)} \frac{e(\rho^\vee)A(np(\rho))}{A(\rho^\vee)}^{e-1}, \end{equation}

in R(T). Write

$$ \begin{aligned} \prod_{i=1}^e\frac{ e(\rho^\vee)A(np(\mu_i)) }{A(\rho^\vee)} &= \prod_{i=1}^e \bigg( W(np(\mu_i)) - \bigg(W(np(\mu_i)) - \frac{ e(\rho^\vee)A(np(\mu_i)) }{A(\rho^\vee)} \bigg) \bigg) \\ &= \prod_{i=1}^e ( W(np(\mu_i))) + C_{\mu,n}\end{aligned} $$

for $C_{\mu,n} \in R(T)$ . Lemma 13.1 ensures that $(W(np(\mu_i)) - ({ e(\rho^\vee)A(np(\mu_i)) }/{A(\rho^\vee)}) )$ is polynomial in n of degree $<|R^+|$ and so, since $W(np(\mu_i))$ has dimension polynomial in n of degree $|R_+^\vee|$ , it follows that $C_\mu = (C_{\mu,n})_{n \geq 0}$ is polynomial of degree $<e|R_+^\vee|$ . Similarly, each

$$ \frac{e(\rho^\vee)A(np(\lambda+\rho))}{A(\rho^\vee)} \frac{e(\rho^\vee)A(np(\rho))}{A(\rho^\vee)}^{e-1} = W(np(\lambda+\rho)) W(np(\rho))^{e-1} + C_{\lambda,n} $$

with $C_\lambda = (C_{\lambda,n})_{n \geq 0}$ polynomial of degree $< e|R_+^\vee|$ . Combining these observations with (13.3) gives that

$$ \prod_{i=1}^e W(p(n\mu_i)) - \sum_{\lambda} m_\lambda W(p(n(\lambda+\rho))) W(p(n\rho))^{e-1} $$

is polynomial of degree $<e|R_+^\vee|$ as desired.