Draft 1 of Talk

The paper: \(T{\hbox{-}}\)equivariant \({\mathsf{K}}{\hbox{-}}\)theory of generalized flag varieties. Kostant-Kumar, 1987.

Notation and setup:

• Everything is over \({\mathbb{C}}\).
• \(A \in \operatorname{Mat}(\ell\times \ell; {\mathbb{Z}})\) a Cartan matrix, \({\mathfrak{g}}= {\mathfrak{g}}(A)\) the associated Kac-Moody Lie algebra, \(G = \operatorname{Lie}(G)\) the associated Kac-Moody group
  • Three types: finite type, affine, indefinite.
  • For finite case: yields a finite dimensional semisimple simply connected algebraic group
• \(T \leq B \leq P \leq G\) with \(G\) a Kac-Moody group, \(T\) a maximal compact torus, \(B\) a Borel and \(P\) a parabolic.
  • In the Lie algebra: \({\mathfrak{h}}\leq {\mathfrak{b}}\leq {\mathfrak{p}}\leq {\mathfrak{g}}\), the Cartan, Borel, parabolic subalgebras.
  • Borel: maximal solvable subalgebra
• \(W\) the Weyl group, a Coxeter or crystallographic group
• \(R(G)\): ?
• \(H^*_T({\operatorname{pt}}; {\mathbb{Z}}) \cong \operatorname{Sym}(T {}^{ \vee }) \coloneqq\operatorname{Sym}(\mathop{\mathrm{Hom}}_{{\mathsf{Alg}}{\mathsf{Grp}}}(T, {\mathbb{C}}^{\times}))\) are symmetric polynomials on the weight lattice.
• $H^T(G/B) = \bigoplus{S\leq G/B} H^_T({\operatorname{pt}}) { \left[ {S} \right] } $ is free over homology of the point with basis \(S\) Schubert classes.
  • Admits a Borel presentation: \begin{align*} H_T^*({\operatorname{pt}}){ {}^{ \scriptstyle\otimes_{H^*_T({\operatorname{pt}})^W }^{2} } } &\to H_T^*(G/B) \\ \lambda\otimes\mu &\mapsto \lambda c_1({\mathcal{L}}(\lambda)) = c_1(G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_\lambda ) ,\end{align*} mapping to to Borel-Weil line bundle. always a \({\mathbb{Q}}{\hbox{-}}\)isomorphism, a \({\mathbb{Z}}{\hbox{-}}\)isomorphism for \(G = \operatorname{GL}_n\).
• \({\mathsf{K}}_T({\operatorname{pt}}) = \left\{{ \sum_{\lambda \in T {}^{ \vee }} m_\lambda e^{\lambda} }\right\}\) are Laurent series in characters of \(T{\hbox{-}}\)representations.

Facts about categorical \({\mathsf{K}}\) from category \({\mathcal{O}}\):

• \([M] = [N] \in {\mathsf{K}}({\mathcal{O}})\) iff \(M\) and \(N\) have the same composition factor multiplicities.

• If \(M\in {\mathcal{O}}\) then \([M] = \sum_{\lambda \in {\mathfrak{h}} {}^{ \vee }} c_\lambda [L(\lambda)]\) where \(c_\lambda = [M: L( \lambda)]\).

• For a fixed central character \(\chi: Z({\mathfrak{g}})\to {\mathbb{C}}\), fixing a block \({\mathcal{O}}_{\chi}\), there are two bases for \({\mathsf{K}}({\mathcal{O}}_\chi)\): \begin{align*} [M] = \sum_{\lambda {~\mathrel{\Big\vert}~}\chi_\lambda = \chi} c_\lambda [L(\lambda)] && [M] = \sum_{\lambda {~\mathrel{\Big\vert}~}\chi_\lambda = \chi} d_\lambda [\operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}}{\mathbb{C}}_\lambda] ,\end{align*} i.e. simples \(L( \lambda)\) and Vermas \(M(\lambda)\).

  • Here \({\mathbb{C}}_\lambda\) is a 1-dim \({\mathfrak{b}}{\hbox{-}}\)module with a trivial \({\mathfrak{n}}{\hbox{-}}\)action, compare \begin{align*} \operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}}{\mathbb{C}}_{\lambda} \cong U({\mathfrak{g}})\otimes_{U({\mathfrak{b}})} {\mathbb{C}}_{\lambda} \rightleftharpoons {\mathcal{L}}(\lambda) \coloneqq G\overset{\scriptscriptstyle {B} }{\times} {\mathbb{C}}_{- \lambda} .\end{align*}

In the finite case for Kac-Moody groups, define \(X^w \coloneqq\mkern 1.5mu\overline{\mkern-1.5muBwB\mkern-1.5mu}\mkern 1.5mu/B \leq G/B\), yields classes \([X^w]\in H_*^T(G/B)\) which form a basis as an \(H_*^T({\operatorname{pt}}){\hbox{-}}\)module. Have an Alexander pairing \({\left\langle {{-}},~{{-}} \right\rangle}\), equivariant cap product composed with pushforward to a point. Use this to define a dual basis to get a basis in cohomology.

There are two natural bases for homology \({\mathsf{K}}_*(G/B)\), which extend \({\mathsf{K}}_*^T({\operatorname{pt}}){\hbox{-}}\)linearly to bases for \({\mathsf{K}}_*^T(G/B)\):

• Structure sheaves \({\mathcal{O}}_{X^w}\), regular functions on \(X_w\)
• Ideal sheaves \(I_{X^w}\), come from functions on \(X^w\) that vanish on the “boundary” \(\displaystyle\bigcup_{\nu < w} X^\nu\).

The change-of-basis matrix is well-known: \begin{align*} [{\mathcal{O}}_{X^w}] = \sum_{\nu \leq w} [I_{X^\nu}] && [I_{X^w}] = \sum_{\nu \leq w} (-1)^{\ell(w) - \ell(\nu)} [{\mathcal{O}}_{X^\nu}] .\end{align*}

Dualize using Alexander pairing to get bases in cohomology \({\mathsf{K}}^*_T(G/B)\), say \begin{align*} {\mathcal{O}}_{X_w} &\leadsto A_w \\ I_{X_w} &\leadsto B_w .\end{align*} We can then look for structure constants for multiplication: \begin{align*} A_\mu A_\nu = \sum_w a^w_{\mu, \nu} A_w && B_\mu B_\nu = \sum_w b^w_{\mu, \nu} B_w .\end{align*}

Famous open problem: compute these in a way that manifestly shows the structure constants are positive. Known in special cases:

• \(\mu = \nu\)
• \(\mu, \nu \in W^P\) where \(G/P\) is a Grassmannian or 2-step flag manifold
• \(W\) is a free Coxeter group

2021: Goldin-Knutson prove positivity by composing operators in the nil-Hecke algebra. Reproves formulas from AJS-Billey and Graham-Willems.

Goal: understand \(H^*(G/B), H^*_T(G/B)\), and \({\mathsf{K}}_T(G/B)\), plus operators acting on them (\(W{\hbox{-}}\)actions, cup products, Demazure operators, etc) in terms of simpler rings, with nice bases, where the operators act in a controllable way on the bases.

Strategy: proved previously for \(H^*\),

• Find a field with a \(W{\hbox{-}}\)action, \(Q \coloneqq Q({\mathfrak{h}} {}^{ \vee })\), rational functions on \({\mathfrak{h}}\)
• Form \(Q{\sharp}W \coloneqq{\mathbb{Z}}[W] {\sharp}Q\) the Hopf smash product, with underlying space \({\mathbb{Z}}[W]\otimes Q\)
• Find a subring \(R \leq Q {\sharp}{\mathbb{Z}}[W]\), find its dual \(R {}^{ \vee }\), and an \(R{\hbox{-}}\)module structure on \(R {}^{ \vee }\), and reduce to studying this.

Similar strategy here:

• Replace \(Q({\mathfrak{h}} {}^{ \vee })\) with \(Q \coloneqq Q(T) \coloneqq\operatorname{ff}A(T)\)
• Find a subring \(Y \leq Q{\sharp}W\), its dual \(\psi\) with \(Y{\hbox{-}}\)actions corresponding to the \(W\) action and \(\left\{{D_w}\right\}\) operators on \({\mathsf{K}}(G/B)\).
  • Here the \(D_w\) are similar to Demazure operators on \(A(T)\).

Recover known results in finite type case: for \(G\) compact simply connected,

• \({\mathsf{K}}^*(G)\) is torsionfree
• There is a surjection \begin{align*} A(T) \twoheadrightarrow{\mathsf{K}}^*(G/T) .\end{align*}
• There is an isomorphism \begin{align*} A(T){ {}^{ \scriptstyle\otimes_{R(G)}^{2} } } { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_T(G/T) .\end{align*}

Ingredients:

• Segal-Atiyah localization theorem
• Equivariant Thom isomorphism

More setup:

• \(Q \coloneqq Q(T) \coloneqq\operatorname{ff}(A(T))\)
  • \(X(T) \coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Alg}}{\mathsf{Grp}}}(T, {\mathbb{C}}^{\times})\)
  • \(A(T) \coloneqq{\mathbb{Z}}[X(T)]\) the character group algebra
• \(Q_W \coloneqq Q {\sharp}{\mathbb{Z}}[W]\)
  • Then \(Q_W \in {\mathsf{Q}{\hbox{-}}\mathsf{Mod}}\) with a free basis \(\left\{{\delta_w}\right\}_{w\in W}\). Write elements as \(\delta_w q \coloneqq\delta_w \otimes q\).
• Define an involution: \begin{align*} (\delta_w q)^t \coloneqq\delta_{w^{-1}}(wq) .\end{align*}
• Replace \(Q(T)\) with \(A(T)\) and define \(A_W\) similarly.
• Find a second basis:

Setup:

• Define an \(A(T){\hbox{-}}\)dual of \(Y\): \begin{align*} \Psi \coloneqq Y {}^{ \vee }\coloneqq\left\{{\psi \in Q_W {}^{ \vee }{~\mathrel{\Big\vert}~}\psi(Y^t) \subseteq A(T)}\right\} \subseteq \mathop{\mathrm{Hom}}_{A(T)}(Y, A(T)) \\ Q_W {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_Q(Q_W, Q) .\end{align*}

• The \(D_w\) operators on \({\mathsf{K}}_T(G/B)\):

  • Write \begin{align*} G/B = \colim_{n} X_n \coloneqq\colim_n \displaystyle\bigcup_{\ell(w) \leq n}BwB/B, && {\mathsf{K}}_T(G/B) \coloneqq\colim_n {\mathsf{K}}_T(X_n) .\end{align*}
  • Define operators \begin{align*} D_{r_i}(n): {\mathsf{K}}_T(\pi_i^{-1}\pi_i X_n) &{\circlearrowleft}\\ x + H_i(n) y &\mapsto x && \forall x, y\in \pi_i^* {\mathsf{K}}_T(\pi_i X_n) ,\end{align*} where \(\pi_i:G/B\to G/P_i\) for \(P_i\) the minimal parabolic containing the simple reflection \(r_i\) and \(H_i(n)\) is the Hopf bundle. Here we’ve used that \begin{align*} {\mathsf{K}}_T(\pi_i^{-1}\pi_i X_n) = {\mathsf{K}}_T(\pi_i X_n)\left\langle{1, H_i(n)}\right\rangle .\end{align*}
  • These lift to operators \begin{align*} D_{r_i}: {\mathsf{K}}_T(G/B) {\circlearrowleft}&& D_w &\coloneqq D_{r_1}\circ \cdots D_{r_n} ,\end{align*} similar to Demazure operators on \(A(T)\).

• The \(W\) action on \({\mathsf{K}}_T(G/B)\):

  • Use that \(W\curvearrowright G/B \cong K/T\) for \(K\) a unitary form of \(G\) (fixed points of \(x\mapsto (\sigma(x)^t)^{-1}\)?)
  • This action is \(T{\hbox{-}}\)equivariant and so induces an operator on \({\mathsf{K}}_T(G/B)\) and \({\mathsf{K}}(G/B)\).
  • We also call these operators \(D_w\)?

• Defining a localization:

  • Glue the canonical restrictions: \begin{align*} \gamma_n: {\mathsf{K}}_T(X_n) \to {\mathsf{K}}_T(X_n^T) \leadsto {\mathsf{K}}_T(G/B)\to {\mathsf{K}}_T(G/B^T) .\end{align*}
  • Put the discrete topology on \(W\) to produce a homeomorphism \begin{align*} \iota: W \cong N_G(T)/T &\to G/B^T \\ w &\mapsto w^{-1}\operatorname{mod}B .\end{align*}
  • Identify an \(A(T){\hbox{-}}\)subalgebra: \begin{align*} {\mathsf{K}}_T(W) \cong \left\{{f: W\to Q {~\mathrel{\Big\vert}~}\operatorname{im}f \subseteq A(T)}\right\} \leq Q_W {}^{ \vee } .\end{align*}
  • Put together: \begin{align*} \gamma: {\mathsf{K}}_T(G/B)\to {\mathsf{K}}_T(G/B^T) \cong {\mathsf{K}}_T(W) \leq Q_W {}^{ \vee } .\end{align*}
  • Restrict to its image to get \begin{align*} \gamma: {\mathsf{K}}_T(G/B) \twoheadrightarrow\Psi .\end{align*}

They explicitly describe an \(A(T){\hbox{-}}\)basis \(\left\{{b_w}\right\}_{w\in W}\), and mention there is a similar result for general Kac-Moody groups. They briefly mention the basis \(\left\{{{\mathcal{O}}_{X^w}}\right\}\), and explicitly say a change-of-basis operator exists, but no mention of positivity.

Algebraic \({\mathsf{K}}{\hbox{-}}\)theory also proved Poincaré for \(n\geq 5\): there are spaces with the homotopy type of a sphere which are not homeomorphism types of a sphere.

Serre-Swan:

Grothendieck’s definition of \({\mathsf{K}}_0\):

Modern perspective: note \({\mathbb{E}}_\infty\) spaces are commutative monoids in spaces: