On a t-exactness property of the Harish-Chandra transform

For a stack $\mathcal{X}$ defined over $\mathbb{C}$, let $D^{b}(\mathcal{X})$ be the bounded algebraically constructible derived category of sheaves on $\mathcal{X}$ with analytic topology, with coefficients in a ring $\mathbb{k}$. All the stacks we encounter will be quotient stacks of the form $X/H$, where $X$ is a scheme and $H$ is an algebraic group over $\mathbb{C}$. The sheaf theory for such stacks is described in [BL06]. We write $\operatorname{pt}=\operatorname{Spec}(\mathbb{C})$. We write $\underline{\mathbb{k}}_{\mathcal{X}}$ for the constant sheaf on $\mathcal{X}$.

For an algebraic group $H$ acting on a scheme $X$ and its subgroup $H^{\prime}$ let

stand for the forgetful functor, that is the $*$-pullback along the projection $H^{\prime}\backslash X\to H\backslash X$. Let

stand for the $!$ and $*$-pushforward from $H^{\prime}\backslash X$ to $H\backslash X$, respectively. If $H^{\prime}$ is clear from the context, we will omit it from the notation, writing $\operatorname{For}^{H},\operatorname{Av}_{!}^{H},\operatorname{Av}_{*}^{H}$, respectively.

We will adopt the same notation for the right action of the group, writing $\prescript{R}{}{\operatorname{For}},\prescript{R}{}{\operatorname{Av}}$ for the right forgetful and averaging functors, when the emphasis is needed.

Let $G$ be a reductive group, $B$ a fixed Borel subgroup, $U\subset B$ its unipotent radical and $T\subset B$ a maximal torus. The Harish-Chandra transform $\mathfrak{hc}_{!}$ for constructible sheaves was introduced by Lusztig [Lus85] in his study of character sheaves. It is given by a pull-push functor along the horocycle correspondence

It was observed by Ginzburg [Gin89] that it produces objects in the categorical center of the finite Hecke category attached to a reductive group $G$ and its Borel subgroup $B$. This observation has been elaborated upon in various contexts in (among other works) [BFO12], [BN09], [Lus], [Lus15], [HL23], [BITV23].

Another remarkable property of the Harish-Chandra transform on the character sheaves, playing a key role in [BFO12], is that it becomes $t$-exact and Verdier self-dual after composing it with the Radon transform functor $R_{*}$. The latter transform is given by the averaging with respect to the unipotent radical $U^{-}$ of the opposite Borel subgroup $B^{-}$, or, equivalently, a pull-push functor along the following diagram:

For a local system $\mathcal{L}$ of finite order, let $D^{b}_{\mathfrak{C}_{\mathcal{L}}}(G)$ be the derived category of character sheaves with the character $\mathcal{L}$ (see Section 2.2 for the precise definition). Our main result is as follows:

We prove the Theorem 1.2.1 by identifying $R_{*}\circ\mathfrak{hc}_{!}$ with the nearby cycles functor along the Vinberg degeneration. The key idea is to use that the nearby cycles functor commutes with the hyperbolic localization, as shown in [Nak16], [Ric19].

All our methods are geometric and can be restated with the same proofs, up to standard modifications, for $D^{b}(\mathcal{X})$ for stacks $\mathcal{X}$ over $\mathbb{C}$ being replaced by either of the following:

• •

  the bounded derived category of holonomic $D$-modules,

• •

  the bounded constructible derived category of $\overline{\mathbb{Q}}_{\ell}$-sheaves on a stack $\mathcal{X}$ defined over an algebraically closed field of characteristic $p>0$ for primes $p\neq\ell$ as in [LO08].

We expect that the methods of the present paper can be extended to handle the parabolic case. We intend to treat it in the subsequent publication.

The idea to relate the nearby cycles functor along the Vinberg degeneration with the functor $R_{*}\circ\mathfrak{hc}_{!}$ appears already in [BFO12], where an analogue of Theorem 1.2.1 in the setting of D-modules is also proved. The same idea was used in [CY17] to prove the Theorem 1.2.1 for unipotent character sheaves (that is for a trivial local system $\mathcal{L}$), with methods applicable to either of the sheaf-theoretic cases, and also in the more general setting associated to a symmetric pair. It is very plausible that the methods of [CY17] can be adapted to treat the general monodromy $\mathcal{L}$. However, we believe that the systematic use of hyperbolic localization presented in this paper makes for a particularly short and straightforward proof.

We would like to thank Roman Bezrukavnikov, Alexander Yom Din, and Yakov Varshavsky for helpful discussions. The work of R.G. was supported by the EPSRC grant EP/V053787/1. R.G. also acknowledges the hospitality of the Hebrew University of Jerusalem during his postdoctoral fellowship. While working on the project, K. T. was at different times funded by the EPSRC programme grant EP/R034826/1 and by the Deutsche Forschungsgemeinschaft SFB 1624 grant, Projektnummer 506632645.

For a scheme $\widetilde{X}$ and a map $f\colon\widetilde{X}\to\mathbb{A}^{1}$, let

stand for the nearby cycles functor. Here $\times_{\mathbb{A}^{1}}$ denotes the fiber product with respect to embeddings $\operatorname{pt}\to\mathbb{A}^{1}\leftarrow\mathbb{G}_{m}$: the embedding of 0 into $\mathbb{A}^{1}$ and the complementary open embedding, respectively.

Assume that $\mathbb{G}_{m}$ acts on $X$ and let $a\colon\mathbb{G}_{m}\times X\to X$ be the action map. Consider $\widetilde{X}=X\times\mathbb{A}^{1}$. Let $f\colon\widetilde{X}\to\mathbb{A}^{1}$ be the projection to the second factor. Following [CY17], define the limit functor

Let $H$ be an algebraic group acting on $X$, and let us fix a cocharacter $\gamma\colon\mathbb{G}_{m}\to H$. This defines the $\mathbb{G}_{m}$-action on $X$, and we write

for the corresponding limit functor.

Let $G$ be a reductive group and $T$ its maximal torus. Choice of a regular cocharacter $\lambda\colon\mathbb{G}_{m}\to T$ fixes a choice of a pair of opposite Borel subgroups $B,B^{-}\supset T$ for which $\lambda$ is dominant and antidominant, respectively. Let $U$ and $U^{-}$ be the unipotent radicals of $B$ and $B^{-}$, respectively.

Let $W=N_{G}(T)/T$ be the Weyl group of $G$, and let $w_{0}\in W$ be its longest element. We have two projections $q_{B}\colon B\to T$ and $q_{B^{-}}\colon B^{-}\to T$.

Let $K\simeq G$ stand for the diagonal subgroup of $G\times G$, and define three subgroups $K_{0}$, $K_{\infty}$, $K^{\prime}\subset G\times G$ as the following fiber products

with respect to the homomorphisms $q_{B}\colon B\to T$ and $q_{B^{-}}\colon B^{-}\to T$.

Fix a cocharacter of $H=G\times G$ given by

Let $X$ be a scheme equipped with a left action of $G\times G$. As shown in [CY17, Lemma 4.12], the limit functor $\operatorname{\mathbb{L}_{\gamma}}$ associated with this case, can be lifted to either of the following functors:

We collect the required properties of this functor in the following lemma. These follow from the standard properties of the nearby cycles functor, see [SGA06, XIII].

Let $X$ be a scheme with an action of an algebraic torus $T$, and let $a\colon T\times X\to X$, $p\colon T\times X\to X$ be the action and projection maps, respectively. Let $\mathcal{L}$ be a rank 1 local system on $T$. We say that $\mathcal{F}\in D^{b}(X)$ is weakly $\mathcal{L}$-equivariant if there is an isomorphism $a^{*}\mathcal{F}\simeq\mathcal{L}\boxtimes\mathcal{F}$. We denote by $D^{b}_{\mathcal{L}}(X)$ the full idempotent-complete triangulated subcategory of $D^{b}(X)$ generated by weakly $\mathcal{L}$-equivariant complexes. We call the complexes in $D^{b}_{\mathcal{L}}(X)$ $\mathcal{L}$-monodromic complexes or monodromic complexes with generalized monodromy $\mathcal{L}$.

If $T=\mathbb{G}_{m}$ and $\mathcal{L}$ is trivial, we will write $D^{b}_{mon}(X)$ for $D^{b}_{\mathcal{L}}(X)$ and refer to this category as a category of $\mathbb{G}_{m}$-monodromic complexes.

For a scheme $X$ with an action of an algebraic group $H$ which is normalized by the action of $T$, we will write $D^{b}_{\mathcal{L}}(H\backslash X)$ for the full triangulated subcategory of $D^{b}(H\backslash X)$ consisting of objects $\mathcal{F}$ with $\operatorname{For}^{H}\mathcal{F}\in D^{b}_{\mathcal{L}}(X)$.

We call a local system $\mathcal{L}$ on $T$ rational if $\alpha^{*}\mathcal{L}$ is trivial for some surjective homomorphism $\alpha\colon T\to T$.

Let $X=G\times G$, and consider actions of $G\times G$ on $X$ by left and right multiplication. Consider a functor

where the superscript $R$ on averaging and forgetful functor is used to emphasized that these are with respect to the groups acting on the right. The notation $\mathfrak{hc}$ stands for Harish-Chandra, and the functor thus defined is commonly called the Harish-Chandra functor. Let

The torus $T$ acts on $K\backslash X/K^{\prime}$ on the right. Let $\mathcal{L}$ be a rational rank 1 local system on $T$. The derived category of character sheaves with character $\mathcal{L}$, denoted by $D^{b}_{\mathfrak{C}_{\mathcal{L}}}(G)$, is the full triangulated idempotent-complete subcategory of $D^{b}(K\backslash X/K)$ generated by the essential image of $D^{b}_{\mathcal{L}}(K\backslash X/K^{\prime})$ under the functor $\chi$.

Consider the diagram

Here $\alpha\colon\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K_{0}}_{B}\circ\operatorname{\mathbb{L}_{\gamma}}\to\operatorname{\mathbb{L}_{\gamma}}\circ\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K}_{B}$ is the natural transformation given by Lemma 2.1.2.

We will prove the following

We will reduce Theorem 2.3.1 to a similar statement of the form “nearby cycles functor commutes with the averaging functor” with $X/K$ replaced first by $X/K^{\prime}$ and then by $\mathcal{B}\times\mathcal{B}$, where $\mathcal{B}=G/B$ is the flag variety of $G$. Consider the following diagram, which is similar to diagram (5), with $X/K$ replaced by $X/K^{\prime}$.

Here $\alpha^{\prime}\colon\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K_{0}}_{B}\circ\operatorname{\mathbb{L}_{\gamma}}\to\operatorname{\mathbb{L}_{\gamma}}\circ\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K}_{B}$ is again the natural transformation given by Lemma 2.1.2.

We have the following

Let $\mathcal{B}=G/B$ be the flag variety of $G$. Consider the following diagram, which is similar to diagram (5), with $X/K^{\prime}$ replaced by $\mathcal{B}\times\mathcal{B}$.

Here $\alpha^{\prime\prime}\colon\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K_{0}}_{B}\circ\operatorname{\mathbb{L}_{\gamma}}\to\operatorname{\mathbb{L}_{\gamma}}\circ\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K}_{B}$ is again the natural transformation given by Lemma 2.1.2.

The implication Theorem 2.3.4 $\Rightarrow$ Theorem 2.3.1 will be deduced from the diagram chase using the standard properties of the nearby cycles functor.

For unipotent monodromy, Theorem 2.3.5 implies the Theorem 2.3.4 in a similar way.

Proof of Theorem 2.3.4 and a simpler Theorem 2.3.5 will use properties of the hyperbolic localization functor, which we proceed to explain.

We recall some notions from [DG14]. Let $X$ be a scheme and $a\colon\mathbb{G}_{m}\times X\rightarrow X$ be a $\mathbb{G}_{m}$-action. Consider the scheme of fixed points $X^{\mathbb{G}_{m}}$. Let by $X^{+}$ and $X^{-}$ the attracting and the repelling schemes respectively. Consider the diagram

where $i_{\pm}$ is the inclusion and $p_{\pm}$ is taking the limit. Let us define the functors of hyperbolic restriction

Denote by $\operatorname{\mathbb{H}}$ the restriction of $\operatorname{\mathbb{H}}^{\pm}$ to the $\mathbb{G}_{m}$-monodromic subcategory.

For a scheme $\widetilde{X}$ and a map $f\colon\widetilde{X}\to\mathbb{A}^{1}$ one has the nearby cycles functor $\psi_{f}$ as in Subsection 2.1. Let $a^{\prime}\colon\mathbb{G}_{m}\times\widetilde{X}\rightarrow\widetilde{X}$ be an action which commutes with the map $f$ for the trivial action on $\mathbb{A}^{1}$. Consider the fixed points of the action and the map $f_{0}\colon\tilde{X}^{\mathbb{G}_{m}}\rightarrow\mathbb{A}^{1}$. The natural transformations from Lemma 2.1.2 yield natural transformations

Consider the $\mathbb{G}_{m}$-action on $\mathcal{B}\times\mathcal{B}$ given by a cocharacter $(\lambda,\lambda)$ for a dominant $\lambda$. This gives us the functor

Moreover, consider $U$ acting on the right factor of $\mathcal{B}\times\mathcal{B}$. This gives us functors

The following lemma is straightforward, see [CGY19, Sect. 1.4.5].

As a corollary we obtain the following proposition

Consider the $\mathbb{G}_{m}$-action on $X/K^{\prime}$ given by a cocharacter

for a dominant $\lambda$. This gives us the functors of hyperbolic localization

For the rest of this subsection we fix $w_{1},w_{2}$ and shorten $\operatorname{\mathbb{H}}:=\operatorname{\mathbb{H}}_{w_{1},w_{2}}$.

Note that $(X/K^{\prime})^{\mathbb{G}_{m}}=(w_{1}B\times w_{2}B)/K^{\prime}$. Moreover, consider $U$ acting on the right factor of $X/K^{\prime}=G\times G/K^{\prime}$. This gives us functors

The following lemma is straightforward generalization of [CGY19, Sect. 1.4.5].

As a corollary we obtain the following proposition

Informally, one can refer to the proposition above as hyperbolic restriction commutes with Radon transform $R_{!}=\operatorname{Av}^{K^{\prime}}_{B,!}\circ\operatorname{For}^{K_{0}}_{B}$.

In this subsection we give a proof of Theorem 2.3.5. The following Lemma is clear.

Using the lemma above it is enough to check that

is an isomorphism. We have the following chain of isomorphisms

The first map is inverse of the unit transformation, see Proposition 3.3.2(b). The second isomorphism is given by Lemma 2.1.2. The third isomorphism is Theorem 3.2.1. The last isomorphism comes from the fact that family is $\mathcal{B}^{\mathbb{G}_{m}}\times\mathcal{B}^{\mathbb{G}_{m}}$ is trivial.

Analogously, we have the following chain of isomorphism

The first isomorphism comes from Lemma 2.1.2. The second isomorphism is Theorem 3.2.1. The third map is inverse of the unit transformation, see Proposition 3.3.2(a). The last isomorphism comes from the fact that family is $\mathcal{B}^{\mathbb{G}_{m}}\times\mathcal{B}^{\mathbb{G}_{m}}$ is trivial.

A diagram chase along the base change transformations implies that the natural transformations (11), (12), and (13) form a commutative triangle.

In this subsection we give a proof of Theorem 2.3.4.

Denote $\operatorname{\mathbb{H}_{\bullet}}=\oplus_{w_{1},w_{2}}\operatorname{\mathbb{H}}_{w_{1},w_{2}}$.

Using the lemma above it is enough to check that

is an isomorphism. We have the following chain of isomorphisms