\include{preamble}

\begin{document}

Speaker: Dinakar Muthiah

Title: Double MV Cycles, the Naito--Sagaki--Saito Crystal, and a view towards affine MV polytopes I, II

February 21--22, 2012

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Notation:
\begin{compactitem}
\item $\fg$ is a symmetrizable Kac--Moody Lie algebra
\item $\Lambda$ is the weight lattice
\item $I$ is the nodes of the Dynkin diagram
\end{compactitem}

The data of a crystal $(\tilde{e}_i, \tilde{f}_i, {\rm wt}, (\eps_i, \phi_i))$.

$B(\lambda)$ is the crystal of the irreducible representation $V(\lambda)$ and $B(\infty)$ is the crystal of $U^-(\fg)$.

We will focus on three ways to construct crystals in the case of finite-dimensional $\fg$: geometric (MV cycles), combinatorial (MV polytopes), algebraic (canonical bases).

\section{Mirkovic--Vilonen (MV) cycles.}

Notation:
\begin{compactitem}
\item $G$ is a simple, simply-connected finite type Kac--Moody group (so data of opposing Borel subgroups $B,B^-$ and maximal torus $T$ is fixed)
\item $\cK = \bC(\!(t)\!)$ and $\cO = \bC[\![t]\!]$.
\item $\cG r_G = G(\cK) / G(\cO)$ is the $\bC$-points of an ind-projective ind-finite type ind-scheme
\item $\Lambda = \hom(\bC^*, T)$ embeds into $\cG r_G$ via the composition $\Spec \cK \to \bC^* \to T$. Given $\mu \in \Lambda$, let $t^\mu$ denote its image.
\end{compactitem}

\subsection{Cell decomposition.}

Let $N, N^-$ be the positive / negative unipotent subgroups of $G$. Then $N(\cK)$ and $N^-(\cK)$ act on $\cG r_G$. Define $S^\mu = N(\cK) t^\mu$ and $T^\gamma = N^-(\cK) t^\gamma$. The intersections $S^\mu \cap T^\gamma$ are finite-dimensional and we have an isomorphism $S^\mu \cap T^\gamma \cong S^0 \cap T^{\gamma + \mu}$. The number of irreducible components of $S^0 \cap T^{-\lambda}$ is given by the Kostant partition function evaluated at $\lambda$. 

\begin{theorem}[Braverman--Finkelberg--Gaitsgory]
$\cM\cV = \coprod_\lambda {\rm Irr}(S^0 \cap T^{-\lambda})$ has a geometrically defined crystal structure for $\fg^\vee$ (Langlands dual Lie algebra) and the resulting crystal is $B(\infty)$.
\end{theorem}

\begin{remark}
This is obvious for rank 1. The proof proceeds by bootstrapping from rank 1 by passing to subminimal Levi subgroups.
\end{remark}

\begin{remark}[Mirkovic--Vilonen]
$\coprod_\lambda {\rm Irr}(S^0 \cap T^{-\lambda})$ indexes a basis in $\rU^-(\fg^\vee)$ (geometric Satake).
\end{remark}

Problem: Parametrize the irreducible components of $S^0 \cap T^\gamma$ and give explicit equations for the irreducible components.

\subsection{MV polytopes.}

This definition is originally due to Anderson and the full description was later given by Kamnitzer.

For all $w \in W$, define $S^{\mu_w}_w = N^w(\cK) t^{\mu_w}$ where $N^w(\cK) = wNw^{-1}(\cK)$. Note that $S^{\mu_{w_0}}_{w_0} = T^{\mu_{w_0}}$. Given a set $\{\mu_w\}_{w \in W}$, define $A(\mu_\bullet) = \bigcap_{w \in W} S^{\mu_w}_w$.

Kamnitzer's ansatz: Let $Z \in {\rm Irr}(S^0 \cap T^\gamma)$. For some particular choice of $\mu_\bullet$, $A(\mu_\bullet) \subset Z$ is open dense.

Note that unless the $\mu_\bullet$ form the vertices of a convex polytope, we will have $A(\mu_\bullet) = \emptyset$.

\begin{theorem}[Kamnitzer]
For every MV cycle $Z$, there exists a unique $A(\mu_\bullet) \subset Z$ which is open dense. The $\mu_\bullet$ that arise this way are precisely those satisfying the ``tropical Pl\"ucker relations''.
\end{theorem}

Let $\Gamma$ be the set of chamber weights, which is $\{w \Lambda_i\}_{i \in I, w \in W}$. Define $M_{w \Lambda_i} = \langle \mu_w, w\Lambda_i \rangle$.

\subsection{Total positivity.}

(References: Fomin--Zelevinsky and Berenstein--Zelevinsky)

Consider the functions $\Delta_{w \Lambda_i} \colon N \to \bC$ given by $\Delta_{w \Lambda_i}(n) = \langle v_{w \Lambda_i} n, v_{\Lambda_i}^* \rangle$ where $v_{w \Lambda_i} \in V(\Lambda_i)$ and $*$ means projection to the highest weight line. These are called {\bf generalized minors}.

\begin{remark}
$N_{\ge 0} = \bigcap_{w \Lambda_i \in \Gamma} \Delta_{w \Lambda_i}^{-1}(\bR_{\ge 0})$.
\end{remark}

These functions satisfy certain relations, which are generalized Pl\"ucker relations. For example, when $G = \SL_3$, we get
\[
\Delta_{s_1 \Lambda_1} \Delta_{s_2 \Lambda_2} = \Delta_{\Lambda_1} \Delta_{s_1 s_2 \Lambda_2} + \Delta_{s_2 s_1 \Lambda_1} \Delta_{\Lambda_2}.
\]
The tropical Pl\"ucker relations are obtained by tropicalizing: addition becomes minimum and multiplication becomes addition. The above relation becomes
\[
M_{s_1 \Lambda_1} + M_{s_2\Lambda_2} = \min(M_{\Lambda_1} + M_{s_1s_2\Lambda_2}, M_{s_2s_1 \Lambda_1} + M_{\Lambda_2}).
\]

\subsection{Idea of proof of Kamnitzer's theorem.}

$V \otimes \cK$ has a filtration by $V \otimes t^n \cO$ and ${\rm val}(v) = \max\{n \mid v \in V \otimes t^n \cO\}$. Then we can write $S^{\mu_w}_w = \{g \in \cG r_G \mid {\rm val}(v_{w\Lambda_i} g) = M_{w \Lambda_i} \text{ for all $i \in I$}\}$. Then ${\rm val}(v_{w \Lambda_i} n) \le {\rm val}(\Delta_{w \Lambda_i}(n))$ for $n \in N(\cK)$.

Consider $\{n \in N(\cK) \mid {\rm val}(\Delta_{w \Lambda_i}(n)) = M_{w\Lambda_i}\}$.  The natural map $\pi \colon N(\cK) \to \cG r_G$ factors through $S^0$. Roughly speaking, $\pi$ will map into $A(\mu_\bullet)$ and it is surjective.

The valuation function satsifies ${\rm val}(ab) = {\rm val}(a) + {\rm val}(b)$ and ${\rm val}(a+b) = \min({\rm val}(a), {\rm val}(b))$ for $a,b$ chosen generically. So generically the $M_\bullet$ satisfy the tropical Pl\"ucker relations.

\begin{remark}
Anderson defined MV polytopes to be the image of $\ol{Z}$ (where $Z$ is an MV cycle) under the moment map for $T$-action on $\cG r_G$. Kamnitzer showed that this polytope is the same as the convex hull of $\{\mu_w\}_{w \in W}$.
\end{remark}

The paths from $\mu_e$ to $\mu_{w_0}$ in the 1-skeleton of MV polytopes are parametrized by reduced decompositions of $w_0$. So the edge lengths of any path give PBW elements in $\rU_q(\fg)$. Two such elements are the same at $q=0$ if and only if they come from the edge lengths of an MV polytope. Hence one gets a crystal structure on the MV polytopes.

\begin{theorem}[Kamnizter]
This crystal structure agrees with the one defined on the MV cycles.
\end{theorem}

\section{Double MV cycles.}

Now we will focus on the case of affine Kac--Moody algebras $\fg$. We consider three ways to construct the crystal $B(\infty)$: geometric (double MV cycles), combinatorial (Naito--Sagaki--Saito crystal, Baumann--Kamnitzer--Tingley affine MV polytopes), canonical bases (Beck--Chari--Pressley, Beck--Nakajima).

Notation:
\begin{compactitem}
\item $G$ is an untwisted affine Kac--Moody group
\item $\cG r_G = G(\cK) / G(\cO)$, which is only a set.
\item $S^\gamma, T^\mu$ defined as before. Now $S^0 \cap T^\gamma$ is the $\bC$-points of a finite type scheme due to results of Braverman--Finkelberg--Gaitsgory and Braverman--Kazhdan--Finkelberg.
\item $\cD\cM\cV = \coprod_\gamma {\rm Irr}(S^0 \cap T^\gamma)$
\end{compactitem}

\begin{theorem}[Braverman--Finkelberg--Gaitsgory]
$\cD\cM\cV$ has a crystal structure which turns it into the $B(\infty)$ crystal for $\fg^\vee$.
\end{theorem}

Problem: Explicitly describe the double MV cycles, i.e., give equations and extract interesting combinatorics.

\subsection{First guess: Double MV cycle ansatz.}

First let $G$ be finite-dimensional. Recall we defined constructible functions
\begin{align*}
D_{w \Lambda_i} \colon \cG r_G &\to \bZ\\
[g] &\mapsto {\rm val}(v_{w \Lambda_i} g).
\end{align*}
For $Z \in \cM\cV$, we define $M(Z)_{w\Lambda_i}$ to be the generic value of $D_{w\Lambda_i}$ on $Z$. A rephrasing of Kamnitzer's result is as follows.

\begin{theorem}[Kamnitzer]
$M(-)_\bullet$ distinguish MV cycles.
\end{theorem}

Now consider the affine case. We define constructible functions
\begin{align*}
D_{\pm w \Lambda_i} \colon S^0 \cap T^\gamma &\to \bZ\\
[n] &\mapsto {\rm val}(v_{\pm w\Lambda_i} n).
\end{align*}
Given $Z \in \cD\cM\cV$, set $M(Z)_{\pm w\Lambda_i}$ to be the generic value of $D_{\pm w\Lambda_i}$ on $Z$. However, these functions no longer distinguish double MV cycles.

\subsection{Type A trick.}

For $G = \widehat{\SL}_n$, we have an embedding $\widehat{\SL}_n \subset \tilde{\GL}_\infty$, and the latter has an action on the Fock space $F$. Fock space has a basis indexed by semi-infinite wedge powers / Maya diagrams / charged partitions. Let $\Gamma$ be any of these index sets. We can also interpret $\Gamma$ as the set of chamber weights for $\tilde{\GL}_\infty$. Given $\gamma \in \Gamma$, define
\begin{align*}
D_\gamma \colon S^0 \cap T^\gamma &\to \bZ\\
[n] &\mapsto {\rm val}(v_\gamma n).
\end{align*}
For $Z \in \cD\cM\cV$, define $M(Z)_\gamma$ to be the generic value of $D_\gamma$ on $Z$.

\begin{theorem}[Muthiah]
$\{M(-)_\gamma\}_{\gamma \in \Gamma}$ distinguish the double MV cycles.
\end{theorem}

Let $\{M_\gamma\}_{\gamma \in \Gamma}$ be a set of integers, which we call pre-NSS data. After imposing some conditions, one constructs an NSS crystal.

\begin{theorem}[Naito--Sagaki--Saito]
The NSS crystal is $B(\infty)$ for $\widehat{\SL}_n$.
\end{theorem}

\begin{theorem}[Baumann--Gaussent, Muthiah]
Let $Z$ be either a double MV cycle or MV cycle. Then $\tilde{f}^k_i Z$ contains a dense open subset of elements of the form $\{x_i (p t^{\phi_i(Z)}) z \mid z \in Z,\ p \in \bC[t^{-1}]^t_k\}$, where $x_i \colon \bG_a \to N$, $\phi_i(Z)$ is the $i$-string function, and $\bC[t^{-1}]^t_k = \{ a_kt^{-k} + \cdots + a_1t^{-1} \mid a_i \in \bC\}$.
\end{theorem}

So given $Z \in \cD\cM\cV$, we have 
\[
M(\tilde{f}_i Z)_\gamma = (\tilde{f}_i M(Z))_\gamma
\]
where the second $\tilde{f}_i$ is the NSS crystal operator.

\begin{theorem}[Muthiah]
$M \colon \cD\cM\cV \to {\rm NSS}$ is an isomorphism of crystals. In particular, this gives a geometric proof that ${\rm NSS}$ is the $B(\infty)$ crystal.
\end{theorem}

\subsection{Open problems.}

What about other types?

Back to the ansatz: we were hoping that the functions $M(Z)_{\pm w\Lambda_i}$ would distinguish double MV cycles. These give vertices of a polytope $\mu^{\pm}_w = \sum_i M(Z)_{\pm w\Lambda_i} w \alpha_i^\vee$. Just as in the finite-dimensional case, the directions of the edges have some constraints and they have lengths $n_i$. But a new phenomena occurs: some of the edges will be multiples of the imaginary root $\delta$.

The number of double MV cycles with a given data appears to be the number of partitions of $n$, which suggests that the additional data we should keep track of is a partition of $n$.

The construction of Baumann--Kamnitzer--Tingley gives exactly these types of pictures via affine MV polytopes. This construction comes from moduli of preprojective algebra representations.

\subsection{Affine PBW bases.}

(References: Beck--Chari--Pressley, Beck--Nakajima)

Start with a Lusztig data: a choice of integer for each positive root that is almost always zero. There is a mysterious operator $S^\lambda$, which one can use together with the braid operators to construct PBW basis elements.

This gives an alternate construction of polytopes by considering paths in the 1-skeleton.

\subsection{Works in progress.}

Baumann--Dunlap--Kamnitzer--Tingley: explicitly describe MV polytopes for $\widehat{\SL}_2$ (combinatorial, no connection to affine PBW bases)

Muthiah--Tingley: Showing that the combinatorial $\widehat{\SL}_2$ polytopes are the same as PBW $\widehat{\SL}_2$ polytopes.

\end{document}