Setup and Goals
The paper: \(T{\hbox{-}}\)equivariant \({\mathsf{K}}{\hbox{-}}\)theory of generalized flag varieties. Kostant-Kumar, 1987.
Notation and setup:
- Everything is over \({\mathbb{C}}\).
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\(A \in \operatorname{Mat}(\ell\times \ell; {\mathbb{Z}})\) 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
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\(T \leq B \leq P \leq G\) 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.
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$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.
Aside: Multiplicities
Facts about categorical \({\mathsf{K}}\) from category \({\mathcal{O}}\):
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\([M] = [N] \in {\mathsf{K}}({\mathcal{O}})\) iff \(M\) and \(N\) have the same composition factor multiplicities.
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If \(M\in {\mathcal{O}}\) then \([M] = \sum_{\lambda \in {\mathfrak{h}} {}^{ \vee }} c_\lambda [L(\lambda)]\) where \(c_\lambda = [M: L( \lambda)]\).
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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.
Kostant-Kumar Paper
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 \(Q \coloneqq Q(T) \coloneqq\operatorname{ff}A(T)\)
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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
Main theorem 1: Defining \(Y\) and its structure theorem
More setup:
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\(Q \coloneqq Q(T) \coloneqq\operatorname{ff}(A(T))\)
- \(X(T) \coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Alg}}{\mathsf{Grp}}}(T, {\mathbb{C}}^{\times})\)
- \(A(T) \coloneqq{\mathbb{Z}}[X(T)]\) the character group algebra
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\(Q_W \coloneqq Q {\sharp}{\mathbb{Z}}[W]\)
- 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:
The elements \(\left\{{y_w }\right\}_{w\in W}\) form a \(Q{\hbox{-}}\)module basis of \(Q_W\): \begin{align*} y_i \coloneqq{\delta_e - e^{-\alpha_i}\delta_{r_i} \over 1 - e^{- \alpha_i}} && e^{\alpha_i}\in X(T) \end{align*} where the character corresponds to the simple root \(\alpha_i\) associated with the reflection \(r_i\). Then define \begin{align*} w = \prod_k r_{i_k} \implies y_w \coloneqq\prod_{k} y_{i_k} .\end{align*}
Moreover, writing \begin{align*} \delta_{\mu ^{-1}} = \sum_{\nu \in W } \ell_{\nu, \mu}y_{\nu^{-1}} && L \coloneqq(\ell_{\nu, \mu})_{\nu, \mu \in W} \end{align*} defines an invertible upper triangular change-of-basis operator with nonzero diagonal.
Define a subring \(Y\) where \(A_W \leq Y \leq Q_W\) by \begin{align*} Y\coloneqq\left\{{y\in Q_W {~\mathrel{\Big\vert}~}y\cdot A(T) \subseteq A(T)}\right\} .\end{align*}
Then \(Y \in {\mathsf{A(T)}{\hbox{-}}\mathsf{Mod}}\) and is free with a basis \(\left\{{y_w }\right\}_{w\in W}\), and is finitely generated as a \({\mathbb{Z}}{\hbox{-}}\)algebra/ring.
Main theorem 2: identifying \({\mathsf{K}}_T(G/B)\)
Setup:
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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*}
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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 \(A(T)\).
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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 \(D_w\)?
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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*}
Let \(G\) be an arbitrary not necessarily symmetrizable Kac-Moody group and \(B\leq G\) a Borel. Then there is an isomorphism \begin{align*} \gamma: {\mathsf{K}}_T(G/B) { { \, \xrightarrow{\sim}\, }}\Psi && \in {\mathsf{Alg}}_{/ {A(T)}} .\end{align*}
There are also maps of operators:
- \(w\in W \mapsto \delta_w\)
- \(D_w \mapsto y_w\).
This induces a commutative diagram
Moreover, \(\gamma_1\) is an isomorphism of \({\mathbb{Z}}{\hbox{-}}\)algebras, and the operators map as
- \(w\in W \mapsto 1\otimes\delta_w\)
- \(D_w\mapsto 1\otimes y_w\)
Replace \(G\) by a maximal compact subgroup containing \(T\) and assume we’re in the finite case. There is an isomorphism \begin{align*} R(T) \otimes_{R(G)} R(T) { { \, \xrightarrow{\sim}\, }}{\mathsf{K}}_T(G/T) ,\end{align*} and \begin{align*} {\mathsf{K}}^*(G) \cong \bigwedge\nolimits^* M && M\in {\mathsf{Z}{\hbox{-}}\mathsf{Mod}} \text{ free, } \operatorname{rank}_{\mathbb{Z}}(M) = \operatorname{rank}G ,\end{align*} i.e. it is an exterior algebra of a free \({\mathbb{Z}}{\hbox{-}}\)module.
Here \begin{align*} R(G) = (({\mathsf{G}{\hbox{-}}\mathsf{Mod}}^{\cong, {\mathrm{fd}}}, \oplus )^{\mathrm{grp}}, \otimes) .\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.
Aside: Interesting Results in \({\mathsf{K}}{\hbox{-}}\)theory
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:
Writing the class number as \({ \operatorname{cl}} ({\mathbb{Q}}(\zeta_p)) = h_1 + h_2\), which measures the extent to which unique factorization fails, if \(p\notdivides h_2\), then Fermat’s last theorem holds for exponent \(p\): \begin{align*} X\coloneqq V(x^n + y^n - z^n) \implies X({\mathbb{Z}})\neq \emptyset .\end{align*}
\({\mathsf{K}}_{4k}({\mathbb{Z}}) = 0 \iff\) the Kummer-Vandiver conjecture