(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
\(G = \mathrm{GL}_n(\mathbb{C}), \mathrm{SL}_n(\mathbb{C}), (\mathbb{C}^*)^n\), 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 \(\mathbb{G}_a \to \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \mathrel{\circlearrowright}\mathbb{C}^2\)
is a nontrivial extension of \(\mathbb{C}\) by itself.
\(T\subset G\) matimal toraus \(T\cong (\mathbb{C}^*)^n\subset \mathrm{GL}_n(\mathbb{C})\)
representation \(G\to \mathrm{GL}(V)\) for \(V\) finite dimensional \(\mathbb{C}\)-vector space.
The set of characters \(X^*(T) := \{ T\to \mathbb{G}_m \} \cong \mathbb{Z}^{\mathrm{rank}G}\)
The set of cocharacters \(X_*(T):= \{\lambda : \mathbb{G}\to T\} \cong \mathbb{Z}^{\mathrm{rank}G}\)
There is a natural pairing
\[ \alpha\circ \lambda : \mathbb{G}_m\to \mathbb{G}_m, \quad \zeta\mapsto \zeta^{\langle \alpha,\lambda\rangle}\]
where \(\langle,\rangle : X^*(T)\times X_*(T) \to \mathbb{Z}\).
\[T\subset G\mathrel{\circlearrowright}V\]
\[V = \bigoplus_{\alpha\in X^*(T)} V_{\alpha}, \quad V_\alpha = \{v| tv = \alpha(t) v\}\]
\[W(V) = \{\alpha\in X^*(T) : V_\alpha\neq 0\}\]
Consider the action of \(\mathrm{GL}_2(\mathbb{C})\mathrel{\circlearrowright}\mathbb{C}^2\), the maximal torus acts
\[(t_1,t_2)e_1 = t_1e_1\] \[(t_1,t_2)e_2 = t_2e_2\]
thich means \(\mathbb{C}^2 = \mathbb{C}e_1 + \mathbb{C}e_2\) are weight spaces of weight \((1,0), (0,1)\).
Symmetric representation
\(\dim V_\alpha = \dim V_{-\alpha} \Leftrightarrow V\cong V^*\)
Example
\(V\oplus V^*\)
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.
\[H^*_G(V) \cong H^*_G(pt) = H^*(BG)\]
to compute it find \(EG\) with a free \(G\) action , then send \(BG = EG/G\).
Example
\(G=\mathbb{C}^*\), \(\mathbb{C}^*\mathrel{\circlearrowright}\mathbb{C}^\infty-\{0\} = EG\), so \(H^*_{\mathbb{C}^*}(pt) = H^*(\mathbb{P}^\infty)\cong \mathbb{Q}[x]\).
For \(G=(\mathbb{C}^*)^m\) we have \(H^*_{(\mathbb{C}^*)^m}(pt) = \mathbb{Q}[x_1,\dots,x_n]\).
\(G\supset T\cong (\mathbb{C}^*)^m\), \(H^*_G(pt) = H^*_T(pt)^W\)
For example \(G = \mathrm{GL}_2, W = S_2\),
\[ H^*_{\mathrm{GL}_2}(pt) = \mathbb{Q}[x_1+x_2,x_1x_2] \]
In general, \(H^*_G(pt)\) 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 \(P_0(V)\subset H^*_G(V)\) depending on \(V\) (while \(H^*_G(V)\) 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 \(V\).
\(\mathbb{C}^*\mathrel{\circlearrowright}\mathbb{C}^N\) 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 \(\mathrm{Spec}(\mathbb{C}[V]^G) = \mathrm{Spec}(\mathbb{C})\).
\(\mathbb{C}^*\mathrel{\circlearrowright}\mathbb{C}^2\) by \(t\cdot (x,y) = (tx, t^{-1}y)\), classical example, whose closed orbits are hyperbolas \(xy=\lambda, \lambda\neq 0\). and the origin.
The invariants are \(\mathbb{C}[xy]\), which is a polynomial ring in 1 variable. This action gives
\[\mathbb{C}^2/\!\!/\mathbb{C}^* = \mathbb{C}\]
\(\mathfrak{g}/\!\!/G\cong(\text{Chevalley}) \mathfrak{t}/\!\!/W \cong \mathbb{C}^{\mathrm{rank}G}\) 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.
\[ \{\text{cohomological integrality}\} \Rightarrow^{\text{Conj}} \{\text{Algorithms for } \mathbb{H}^*(V/\!\!/G)\}\]
where \(\mathbb{H}\) stands for intersection cohomology.
Here
\[ \mathbb{H}^*(X) = \begin{cases} H^*(X) & \text{if } X \text{ is smooth} \\ \text{some other cohomology encoding singularities} & \text{if } X \text{ is singular} \end{cases}\]
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 \(G\)
\(G = \mathrm{GL}_n(\mathbb{C}), V = \mathbb{C}^n \oplus (\mathbb{C}^n)^* = T^*\mathbb{C}^n\)
\(\lambda: \mathbb{G}_m\to G\)
\(\lambda(t) = \begin{pmatrix} t^2 & & & & & \\ & t^2 & & & & \\ & & t & & & \\ & & & t & & \\ & & & & 1 & \\ & & & & & 1 \end{pmatrix}\)
\(G^\lambda = \{g | \lambda(t)g\lambda(t)^{-1} = g\}\)
\(G^\lambda = \begin{pmatrix} (\dots) & & \\ & (\dots) & \\ & & (\dots) \end{pmatrix}\) (a three block matrix)
\(V^\lambda = \{v | \lambda(t)v = v\}\), a representation of \(G^\lambda\)
\(V^\lambda = T^*\begin{pmatrix} 0 \\ 0 \\ * \\ \end{pmatrix}\)
$ G^{} = {g | _{t} (t)g(t)^{-1} }G$
\(\mathrm{GL}_n^{\lambda\ge 0} = \begin{pmatrix} * & * & * \\ & * & * \\ & & * \end{pmatrix}\)
is the parabolic subgroup of \(G\) corresponding to \(\lambda\).
$ G^{}V^{} = {v | _{t} (t)v }$
\(V^{\lambda\ge 0} = \mathbb{C}^n \oplus \begin{pmatrix} 0 \\ 0 \\ * \end{pmatrix}\)
Induction Diagram
The induction map is
\[\mathrm{Ind}_\lambda = (p_\lambda)_* (q_\lambda)^* : H^*_{G^{\lambda}}(V^\lambda) \to H^*_G(V)\]
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 \(\alpha : T\to \mathbb{C}^*\)
\[\,\mathrm{d}\alpha(1) : \mathfrak{t}= \mathrm{Lie}(T)\to \mathbb{C}\in \mathfrak{t}^* \subset H^*_T(pt) \cong \mathrm{Sym}(\mathfrak{t}^*)\]
\[k_\lambda = \frac{\prod_{\alpha\in W(V), \langle \alpha,\lambda\rangle > 0} \alpha^{\dim V_\alpha}}{\prod_{\alpha\in W(\mathfrak{g}), \langle \alpha,\lambda\rangle > 0} \alpha^{\dim \mathfrak{g}_\alpha}} \in \mathrm{Frac}(H^*_T(pt))\]
\[\mathrm{Ind}_\lambda = \frac{1}{|W^\lambda|}\sum_{w\in W} w\cdot (f\cdot k_\lambda) \in H^*_G(V)\]
where \(f\in H^*_{G^\lambda}(V^\lambda)\).
A few more things
On \(X_*(T)\) we define a equivalence relation:
\[\lambda\sim \mu \Leftrightarrow\begin{cases} G^\lambda = G^\mu \\ V^\lambda = V^\mu \end{cases}\]
\[W\mathrel{\circlearrowright}P_v = X_*(T)/\sim\] is a finite set.
For \(\lambda\in X_*(T)\), \(G_\lambda = \ker(G^\lambda \to \mathrm{GL}(V^\lambda))\cap Z(G^\lambda)\).
\(W_\lambda = \{ w\in W | \overline{w\cdot \lambda} = \overline{\lambda} \text{ in } P_v\} \subset W\)
\(\varepsilon_{v,\lambda} : W_\lambda \to \{\pm 1\}\) is characterized by
\[ w\cdot k_\lambda = \varepsilon_{v,\lambda}(w) k_\lambda\]
Main Theorem
Let \(V\) be a symmetric represetnation of reductive \(G\).
Then there exists finite dimensional subspaces \(P^\lambda\subset H^*_{G^\lambda}(V^\lambda)\), graded and stable under \(W^\lambda\) action, such that
\[ \bigoplus_{\tilde{\lambda}\in P_v/W} (P_\lambda \otimes H^*_{G_\lambda}(pt))^{\varepsilon_{v,\lambda}} \xrightarrow{\bigoplus_{\tilde{\lambda}} \mathrm{Ind}_\lambda} H^*_G(V)\]
The left side is isotypic component for \(W^\lambda\)-action.