Author: Eiko

Time: 2024-10-08 10:44:24 - 2024-10-08 10:44:24 (UTC)

(seminar notes)

Cohomological Integrality For Symmetric Quotient Stacks

arXiv: 2406.09218 2408.15786

M.Reineke Donaldson-Thomas (DT) Invariants

Bettei Title: DT invariants of symmetric representations of reductive groups

We work over $\mathbb{C}$.

Situation

$G={\mathrm{GL}}_n(\mathbb{C}),{\mathrm{SL}}_n(\mathbb{C}),({\mathbb{C}}^{\ast }{)}^n$, more generally, $G$ reductive, i.e. unipotent radical is trivial)

Which is equivalent to linearly reductive (all finite dimension representation are semisimple).

A non-example is ${\mathbb{G}}_a\to \left(\begin{array}{cc}1& \ast \\ 0& 1\end{array}\right)\circlearrowright {\mathbb{C}}^2$

is a nontrivial extension of $\mathbb{C}$ by itself.

• $T\subset G$ matimal toraus $T\cong ({\mathbb{C}}^{\ast }{)}^n\subset {\mathrm{GL}}_n(\mathbb{C})$

• representation $G\to \mathrm{GL}(V)$ for $V$ finite dimensional $\mathbb{C}$-vector space.

• The set of characters $X^{\ast }(T):=\{T\to {\mathbb{G}}_m\}\cong {\mathbb{Z}}^{\mathrm{rank}G}$

• The set of cocharacters $X_{\ast }(T):=\{\lambda :\mathbb{G}\to T\}\cong {\mathbb{Z}}^{\mathrm{rank}G}$

There is a natural pairing

$$\alpha \circ \lambda :{\mathbb{G}}_m\to {\mathbb{G}}_m,\zeta \mapsto {\zeta }^{⟨\alpha ,\lambda ⟩}$$

where $⟨,⟩:X^{\ast }(T)\times X_{\ast }(T)\to \mathbb{Z}$.

$$T\subset G\circlearrowright V$$

$$V=\underset{\alpha \in X^{\ast }(T)}{⨁}{V}_{\alpha },V_{\alpha }=\{v|tv=\alpha (t)v\}$$

$$W(V)=\{\alpha \in X^{\ast }(T):V_{\alpha }\ne 0\}$$

Consider the action of ${\mathrm{GL}}_2(\mathbb{C})\circlearrowright {\mathbb{C}}^2$, the maximal torus acts

$$(t_1,t_2)e_1=t_1{e}_1$$ $$(t_1,t_2)e_2=t_2{e}_2$$

thich means ${\mathbb{C}}^2=\mathbb{C}{e}_1+\mathbb{C}{e}_2$ are weight spaces of weight $(1,0),(0,1)$.

Symmetric representation

$\dim V_{\alpha }=\dim V_{-\alpha }\iff V\cong V^{\ast }$

Example

• $V\oplus V^{\ast }$

• Any representation of ${\mathrm{SL}}_2(\mathbb{C})$

• $\mathfrak{g}=\mathrm{Lie}(G)$ adjoint representation (Killing form)

Weil Groups

By definition, $W=N_G(T)/T$ where $N_G(T)=\{g|gT=Tg\}$ is the normalizer.

For example $G={\mathrm{GL}}_n$, $W_{{\mathrm{GL}}_n}=S_n$ is the symmetric group.

$W$ acts on $W(V)$.

Cohomological Integrality

Setup: $V$ is a finite dimensional $G$ representation.

$$H_G^{\ast }(V)\cong H_G^{\ast }(pt)=H^{\ast }(BG)$$

to compute it find $EG$ with a free $G$ action , then send $BG=EG/G$.

Example

$G={\mathbb{C}}^{\ast }$, ${\mathbb{C}}^{\ast }\circlearrowright {\mathbb{C}}^{\mathrm{\infty }}-\{0\}=EG$, so $H_{{\mathbb{C}}^{\ast }}^{\ast }(pt)=H^{\ast }({\mathbb{P}}^{\mathrm{\infty }})\cong \mathbb{Q}[x]$.

For $G=({\mathbb{C}}^{\ast }{)}^m$ we have $H_{({\mathbb{C}}^{\ast }{)}^m}^{\ast }(pt)=\mathbb{Q}[x_1,\dots ,x_n]$.

$G\supset T\cong ({\mathbb{C}}^{\ast }{)}^m$, $H_G^{\ast }(pt)=H_T^{\ast }(pt{)}^W$

For example $G={\mathrm{GL}}_2,W=S_2$,

$$H_{{\mathrm{GL}}_2}^{\ast }(pt)=\mathbb{Q}[x_1+x_2,x_1{x}_2]$$

In general, $H_G^{\ast }(pt)$ is always a polynomial ring $\mathbb{Q}[y_1,\dots ,y_{\mathrm{rank}G}]$.

Goal of Cohomological Integrality

is to extract a finite dimensional subspace $P_0(V)\subset H_G^{\ast }(V)$ depending on $V$ (while $H_G^{\ast }(V)$ is infinite dimensional and does not depend on $V$), that generates in the sense of parabolic induction

\[P_0 = \text{unipotent cohomology of }V/G = \begin{cases} \text{character sheaves} & Lusztig \text{Hecke eigensheaves & Geometric Langlands \end{cases} \]

Context and Motivation

• Topology of $(V,G)$ , also $V//G=\mathrm{Spec}(\mathbb{C}[V{]}^G)$.

  The ring is finitely generated by Hilbert’s theorem. $V//G$ classifies closed $G$-orbits in $V$.

• ${\mathbb{C}}^{\ast }\circlearrowright {\mathbb{C}}^N$ by rescaling, then everything contracts to the origin, so there is only one closed orbit (the origin). i.e. $V//G=\{0\}$.

  In terms of invariants, there are no invariants under scaling except constants, which match our expectation since $\mathrm{Spec}(\mathbb{C}[V{]}^G)=\mathrm{Spec}(\mathbb{C})$.

• ${\mathbb{C}}^{\ast }\circlearrowright {\mathbb{C}}^2$ by $t\cdot (x,y)=(tx,t^{-1}y)$, classical example, whose closed orbits are hyperbolas $xy=\lambda ,\lambda \ne 0$. and the origin.

  The invariants are $\mathbb{C}[xy]$, which is a polynomial ring in 1 variable. This action gives

  $${\mathbb{C}}^2//{\mathbb{C}}^{\ast }=\mathbb{C}$$

• $\mathfrak{g}//G\cong (\text{Chevalley})\mathfrak{t}//W\cong {\mathbb{C}}^{\mathrm{rank}G}$ where $\mathfrak{t}=Lie(T)$.

Computing generators of $\mathbb{C}[V{]}^G$ is very difficult, even for $G={\mathrm{SL}}_2(\mathbb{C})$, which are polynomial invariants of binary forms.

But we can try to understand the topology of $V//G$. Here comes the cohomological integrality.

$$\{\text{cohomological integrality}\}{\Rightarrow }^{\text{Conj}}\{\text{Algorithms for }{\mathbb{H}}^{\ast }(V//G)\}$$

where $\mathbb{H}$ stands for intersection cohomology.

Here

$${\mathbb{H}}^{\ast }(X)=\{\begin{array}{ll}{H}^{\ast }(X)& \text{if }X\text{ is smooth}\\ \text{some other cohomology encoding singularities}& \text{if }X\text{ is singular}\end{array}$$

The topology of $\mathfrak{M}$ a smooth artin stack $\to \mathcal{M}$ a good moduli space.

Stacks can fully encode equivariant cohomologies.

Introduce and new enumerative invariants fo $G,V$.

Operations

• $V$ a representation of $G$

  $G={\mathrm{GL}}_n(\mathbb{C}),V={\mathbb{C}}^n\oplus ({\mathbb{C}}^n{)}^{\ast }=T^{\ast }{\mathbb{C}}^n$

• $\lambda :{\mathbb{G}}_m\to G$

  $\lambda (t)=\left(\begin{array}{cccccc}{t}^2& & & & & \\ & t^2& & & & \\ & & t& & & \\ & & & t& & \\ & & & & 1& \\ & & & & & 1\end{array}\right)$

• $G^{\lambda }=\{g|\lambda (t)g\lambda (t{)}^{-1}=g\}$

  $G^{\lambda }=\left(\begin{array}{ccc}(\dots )& & \\ & (\dots )& \\ & & (\dots )\end{array}\right)$ (a three block matrix)

• $V^{\lambda }=\{v|\lambda (t)v=v\}$, a representation of $G^{\lambda }$

  $V^{\lambda }=T^{\ast }\left(\begin{array}{c}0\\ 0\\ \ast \end{array}\right)$

• $ G^{} = {g | _{t} (t)g(t)^{-1} }G$

  ${\mathrm{GL}}_n^{\lambda \ge 0}=\left(\begin{array}{ccc}\ast & \ast & \ast \\ & \ast & \ast \\ & & \ast \end{array}\right)$

  is the parabolic subgroup of $G$ corresponding to $\lambda$.

• $ G^{}V^{} = {v | _{t} (t)v }$

  $V^{\lambda \ge 0}={\mathbb{C}}^n\oplus \left(\begin{array}{c}0\\ 0\\ \ast \end{array}\right)$

Induction Diagram

The induction map is

$${\mathrm{Ind}}_{\lambda }=(p_{\lambda }{)}_{\ast }(q_{\lambda }{)}^{\ast }:H_{G^{\lambda }}^{\ast }(V^{\lambda })\to H_G^{\ast }(V)$$

i.e.

$$\mathbb{Q}[x_1,\dots ,x_{\mathrm{rank}G}{]}^{W^{\lambda }}\to \mathbb{Q}[x_1,\dots ,x_{\mathrm{rank}G}{]}^W$$

where $W^{\lambda }$ is the Weyl group of $G^{\lambda }$.

Explicit Formula

Let $\alpha :T\to {\mathbb{C}}^{\ast }$

$$\mathrm{d}\alpha (1):\mathfrak{t}=\mathrm{Lie}(T)\to \mathbb{C}\in {\mathfrak{t}}^{\ast }\subset H_T^{\ast }(pt)\cong \mathrm{Sym}({\mathfrak{t}}^{\ast })$$

$$k_{\lambda }=\frac{\prod _{\alpha \in W(V),⟨\alpha ,\lambda ⟩>0}{\alpha }^{\dim V_{\alpha }}}{\prod _{\alpha \in W(\mathfrak{g}),⟨\alpha ,\lambda ⟩>0}{\alpha }^{\dim {\mathfrak{g}}_{\alpha }}}\in \mathrm{Frac}(H_T^{\ast }(pt))$$

$${\mathrm{Ind}}_{\lambda }=\frac{1}{|W^{\lambda }|}\sum _{w\in W}w\cdot (f\cdot k_{\lambda })\in H_G^{\ast }(V)$$

where $f\in H_{G^{\lambda }}^{\ast }(V^{\lambda })$.

A few more things

On $X_{\ast }(T)$ we define a equivalence relation:

$$\lambda \sim \mu \iff \{\begin{array}{l}{G}^{\lambda }=G^{\mu }\\ V^{\lambda }=V^{\mu }\end{array}$$

$$W\circlearrowright P_v=X_{\ast }(T)/\sim$$ is a finite set.

For $\lambda \in X_{\ast }(T)$, $G_{\lambda }=\ker (G^{\lambda }\to \mathrm{GL}(V^{\lambda }))\cap Z(G^{\lambda })$.

$W_{\lambda }=\{w\in W|\overline{w\cdot \lambda }=\bar{\lambda }\text{ in }{P}_v\}\subset W$

${\epsilon }_{v,\lambda }:W_{\lambda }\to \{\pm 1\}$ is characterized by

$$w\cdot k_{\lambda }={\epsilon }_{v,\lambda }(w)k_{\lambda }$$

Main Theorem

Let $V$ be a symmetric represetnation of reductive $G$.

Then there exists finite dimensional subspaces $P^{\lambda }\subset H_{G^{\lambda }}^{\ast }(V^{\lambda })$, graded and stable under $W^{\lambda }$ action, such that

$$\underset{\tilde{\lambda }\in P_v/W}{⨁}(P_{\lambda }\otimes H_{G_{\lambda }}^{\ast }(pt){)}^{{\epsilon }_{v,\lambda }}\stackrel{\underset{\tilde{\lambda }}{⨁}{\mathrm{Ind}}_{\lambda }}{\to }{H}_G^{\ast }(V)$$

The left side is isotypic component for $W^{\lambda }$-action.