--- created: 2021-12-06T23:31 updated: 2023-12-28T09:22 tags: - my/talks --- # Draft 1 of Talk ## Setup and Goals :::{.remark} The paper: $T\dash$equivariant \(\K\dash\)theory of generalized flag varieties. Kostant-Kumar, 1987. ::: :::{.remark} Notation and setup: - Everything is over $\CC$. - $A \in \Mat(\ell\times \ell; \ZZ)$ a Cartan matrix, $\lieg = \lieg(A)$ the associated Kac-Moody Lie algebra, $G = \Lie(G)$ 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: $\lieh\leq \lieb \leq \liep \leq \lieg$, the Cartan, Borel, parabolic subalgebras. - Borel: maximal solvable subalgebra - $W$ the Weyl group, a Coxeter or crystallographic group - $R(G)$: ? - $H^*_T(\pt; \ZZ) \cong \Sym(T\dual) \da \Sym(\Hom_{\Alg\Grp}(T, \CC\units))$ are symmetric polynomials on the weight lattice. - $H^*_T(G/B) = \bigoplus_{S\leq G/B} H^*_T(\pt)\adjoin{S}$ is free over homology of the point with basis $S$ Schubert classes. - Admits a Borel presentation: \[ H_T^*(\pt)\tensorpower{H^*_T(\pt)^W }{2} &\to H_T^*(G/B) \\ \lambda\tensor \mu &\mapsto \lambda c_1(\mcl(\lambda)) = c_1(G\mix{B} \CC_\lambda ) ,\] mapping to to Borel-Weil line bundle. always a $\QQ\dash$isomorphism, a $\ZZ\dash$isomorphism for $G = \GL_n$. - $\K_T(\pt) = \ts{ \sum_{\lambda \in T\dual} m_\lambda e^{\lambda} }$ are Laurent series in characters of $T\dash$representations. ::: ## Aside: Multiplicities :::{.remark} Facts about categorical $\K$ from category $\OO$: - $[M] = [N] \in \K(\OO)$ iff $M$ and $N$ have the same composition factor multiplicities. - If $M\in \OO$ then $[M] = \sum_{\lambda \in \lieh\dual} c_\lambda [L(\lambda)]$ where $c_\lambda = [M: L( \lambda)]$. - For a fixed central character $\chi: Z(\lieg)\to \CC$, fixing a block $\OO_{\chi}$, there are two bases for $\K(\OO_\chi)$: \[ [M] = \sum_{\lambda \st \chi_\lambda = \chi} c_\lambda [L(\lambda)] && [M] = \sum_{\lambda \st \chi_\lambda = \chi} d_\lambda [\Ind_\lieb^\lieg \CC_\lambda] ,\] i.e. simples $L( \lambda)$ and Vermas $M(\lambda)$. - Here $\CC_\lambda$ is a 1-dim $\lieb\dash$module with a trivial $\lien\dash$action, compare \[ \Ind_\lieb^\lieg \CC_{\lambda} \cong U(\lieg)\tensor_{U(\lieb)} \CC_{\lambda} \mapstofrom \mcl(\lambda) \da G\mix{B} \CC_{- \lambda} .\] ::: :::{.remark} In the finite case for Kac-Moody groups, define $X^w \da \bar{BwB}/B \leq G/B$, yields classes $[X^w]\in H_*^T(G/B)$ which form a basis as an $H_*^T(\pt)\dash$module. Have an Alexander pairing $\inner{\wait}{\wait}$, 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** $\K_*(G/B)$, which extend $\K_*^T(\pt)\dash$linearly to bases for $\K_*^T(G/B)$: - Structure sheaves $\OO_{X^w}$, regular functions on $X_w$ - Ideal sheaves $I_{X^w}$, come from functions on $X^w$ that vanish on the "boundary" $\Union_{\nu < w} X^\nu$. The change-of-basis matrix is well-known: \[ [\OO_{X^w}] = \sum_{\nu \leq w} [I_{X^\nu}] && [I_{X^w}] = \sum_{\nu \leq w} (-1)^{\ell(w) - \ell(\nu)} [\OO_{X^\nu}] .\] Dualize using Alexander pairing to get bases in cohomology $\K^*_T(G/B)$, say \[ \OO_{X_w} &\leadsto A_w \\ I_{X_w} &\leadsto B_w .\] We can then look for structure constants for multiplication: \[ A_\mu A_\nu = \sum_w a^w_{\mu, \nu} A_w && B_\mu B_\nu = \sum_w b^w_{\mu, \nu} B_w .\] 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 :::{.remark} Goal: understand $H^*(G/B), H^*_T(G/B)$, and $\K_T(G/B)$, plus operators acting on them ($W\dash$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. ::: :::{.remark} Strategy: proved previously for $H^*$, - Find a field with a $W\dash$action, $Q \da Q(\lieh\dual)$, rational functions on $\lieh$ - Form $Q\size W \da \ZZ[W] \size Q$ the Hopf smash product, with underlying space $\ZZ[W]\tensor Q$ - Find a subring $R \leq Q \size \ZZ[W]$, find its dual $R\dual$, and an $R\dash$module structure on $R\dual$, and reduce to studying this. Similar strategy here: - Replace $Q(\lieh\dual)$ with $Q \da Q(T) \da \ff A(T)$ - Find a subring $Y \leq Q\size W$, its dual $\psi$ with $Y\dash$actions corresponding to the $W$ action and $\ts{D_w}$ operators on $\K(G/B)$. - Here the $D_w$ are similar to Demazure operators on $A(T)$. ::: :::{.remark} Recover known results in finite type case: for $G$ compact simply connected, - $\K^*(G)$ is torsionfree - There is a surjection \[ A(T) \surjects \K^*(G/T) .\] - There is an isomorphism \[ A(T)\tensorpower{R(G)}{2} \iso \K_T(G/T) .\] Ingredients: - Segal-Atiyah localization theorem - Equivariant Thom isomorphism ::: ## Main theorem 1: Defining $Y$ and its structure theorem :::{.remark} More setup: - $Q \da Q(T) \da \ff(A(T))$ - $X(T) \da \Hom_{\Alg\Grp}(T, \CC\units)$ - $A(T) \da \ZZ[X(T)]$ the character group algebra - $Q_W \da Q \size \ZZ[W]$ - Then $Q_W \in \mods{Q}$ with a free basis $\ts{\delta_w}_{w\in W}$. Write elements as $\delta_w q \da \delta_w \tensor q$. - Define an involution: \[ (\delta_w q)^t \da \delta_{w\inv}(wq) .\] - Replace $Q(T)$ with $A(T)$ and define $A_W$ similarly. - Find a second basis: ::: :::{.proposition title="?"} The elements $\ts{y_w }_{w\in W}$ form a $Q\dash$module basis of $Q_W$: \[ y_i \da {\delta_e - e^{-\alpha_i}\delta_{r_i} \over 1 - e^{- \alpha_i}} && e^{\alpha_i}\in X(T) \] where the character corresponds to the simple root $\alpha_i$ associated with the reflection $r_i$. Then define \[ w = \prod_k r_{i_k} \implies y_w \da \prod_{k} y_{i_k} .\] Moreover, writing \[ \delta_{\mu \inv} = \sum_{\nu \in W } \ell_{\nu, \mu}y_{\nu\inv} && L \da (\ell_{\nu, \mu})_{\nu, \mu \in W} \] defines an invertible upper triangular change-of-basis operator with nonzero diagonal. ::: :::{.theorem title="First main theorem"} Define a subring $Y$ where $A_W \leq Y \leq Q_W$ by \[ Y\da \ts{y\in Q_W \st y\cdot A(T) \subseteq A(T)} .\] Then $Y \in \mods{A(T)}$ and is free with a basis $\ts{y_w }_{w\in W}$, and is finitely generated as a $\ZZ\dash$algebra/ring. ::: ## Main theorem 2: identifying $\K_T(G/B)$ :::{.remark} Setup: - Define an $A(T)\dash$dual of $Y$: \[ \Psi \da Y\dual \da \ts{\psi \in Q_W\dual \st \psi(Y^t) \subseteq A(T)} \subseteq \Hom_{A(T)}(Y, A(T)) \\ Q_W \dual \da \Hom_Q(Q_W, Q) .\] - The $D_w$ operators on $\K_T(G/B)$: - Write \[ G/B = \colim_{n} X_n \da \colim_n \Union_{\ell(w) \leq n}BwB/B, && \K_T(G/B) \da \colim_n \K_T(X_n) .\] - Define operators \[ D_{r_i}(n): \K_T(\pi_i\inv \pi_i X_n) &\selfmap \\ x + H_i(n) y &\mapsto x && \forall x, y\in \pi_i^* \K_T(\pi_i X_n) ,\] 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\[ \K_T(\pi_i\inv \pi_i X_n) = \K_T(\pi_i X_n)\gens{1, H_i(n)} .\] - These lift to operators \[ D_{r_i}: \K_T(G/B) \selfmap && D_w &\da D_{r_1}\circ \cdots D_{r_n} ,\] similar to Demazure operators on $A(T)$. - The $W$ action on $\K_T(G/B)$: - Use that $W\actson G/B \cong K/T$ for $K$ a unitary form of $G$ (fixed points of $x\mapsto (\sigma(x)^t)\inv$?) - This action is $T\dash$equivariant and so induces an operator on $\K_T(G/B)$ and $\K(G/B)$. - We also call these operators $D_w$? - Defining a localization: - Glue the canonical restrictions: \[ \gamma_n: \K_T(X_n) \to \K_T(X_n^T) \leadsto \K_T(G/B)\to \K_T(G/B^T) .\] - Put the discrete topology on $W$ to produce a homeomorphism \[ \iota: W \cong N_G(T)/T &\to G/B^T \\ w &\mapsto w\inv \mod B .\] - Identify an $A(T)\dash$subalgebra: \[ \K_T(W) \cong \ts{f: W\to Q \st \im f \subseteq A(T)} \leq Q_W\dual .\] - Put together: \[ \gamma: \K_T(G/B)\to \K_T(G/B^T) \cong \K_T(W) \leq Q_W\dual .\] - Restrict to its image to get \[ \gamma: \K_T(G/B) \surjects \Psi .\] ::: :::{.theorem title="Second main theorem"} Let $G$ be an arbitrary not necessarily symmetrizable Kac-Moody group and $B\leq G$ a Borel. Then there is an isomorphism \[ \gamma: \K_T(G/B) \iso \Psi && \in \Alg\slice{A(T)} .\] There are also maps of operators: - $w\in W \mapsto \delta_w$ - $D_w \mapsto y_w$. ::: :::{.corollary title="Lifting from equivariant K to usual K"} This induces a commutative diagram ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_11-01-18.png) Moreover, $\gamma_1$ is an isomorphism of $\ZZ\dash$algebras, and the operators map as - $w\in W \mapsto 1\tensor \delta_w$ - $D_w\mapsto 1\tensor y_w$ ::: :::{.corollary title="?"} Replace $G$ by a maximal compact subgroup containing $T$ and assume we're in the finite case. There is an isomorphism \[ R(T) \tensor_{R(G)} R(T) \iso \K_T(G/T) ,\] and \[ \K^*(G) \cong \Extalg^* M && M\in \mods{Z} \text{ free, } \rank_\ZZ(M) = \rank G ,\] i.e. it is an exterior algebra of a free $\ZZ\dash$module. Here \[ R(G) = ((\mods{G}^{\cong, \fd}, \oplus )^{\mathrm{grp}}, \tensor) .\] ::: :::{.remark} They explicitly describe an $A(T)\dash$basis $\ts{b_w}_{w\in W}$, and mention there is a similar result for general Kac-Moody groups. They briefly mention the basis $\ts{\OO_{X^w}}$, and explicitly say a change-of-basis operator exists, but no mention of positivity. ::: ## Aside: Interesting Results in \(\K\dash\)theory :::{.remark} Algebraic \(\K\dash\)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. ::: :::{.remark} Serre-Swan: ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-17-59.png) Grothendieck's definition of $\K_0$: ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-22-10.png) Modern perspective: note $\EE_\infty$ spaces are commutative monoids in spaces: ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-33-57.png) ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-23-11.png) ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-23-28.png) ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-24-05.png) ![](Projects/0000%20Talks/Archive/Kac%20Moody/figures/2021-12-07_10-26-17.png) ::: :::{.conjecture title="Kummer-Vandiver"} Writing the class number as $\cl(\QQ(\zeta_p)) = h_1 + h_2$, which measures the extent to which unique factorization fails, if $p\notdivides h_2$, then Fermat's last theorem holds for exponent $p$: \[ X\da V(x^n + y^n - z^n) \implies X(\ZZ)\neq \emptyset .\] ::: :::{.theorem title="?"} $\K_{4k}(\ZZ) = 0 \iff$ the Kummer-Vandiver conjecture :::