\include{preamble}

\begin{document}

Speaker: Tatsuyuki Hikita

Title: Affine Springer fibers of type A and combinatorics of diagonal coinvariants

February 21, 2012

\section{Diagonal coinvariants.}

Set $\bC[\bx, \by] = \bC[x_1, y_1, \dots, x_n, y_n]$ which has a bigrading given by $\deg(x_i) = (1,0)$ and $\deg(y_i) = (0,1)$. This has an action of the symmetric group $\fS_n$. Let $\bC[\bx,\by]_+^{\fS_n}$ be the space of $\fS_n$-invariants without constant term. The {\bf diagonal coinvariants} is the quotient $DR_n = \bC[\bx, \by] / \langle \bC[\bx,\by]_+^{\fS_n} \rangle$. This has a bigrading which is preserved by $\fS_n$.

The {\bf Frobenius characteristic} is the map $\cF$ from ${\rm Rep}(\fS_n)$ to the degree $n$ symmetric functions which sends the irreducible $L_\lambda$ to $s_\lambda$.

Problem: Describe $\cF(DR_n, z; q,t) := \sum_{i,j} t^i q^j \cF((DR_n)_{i,j})$.

\begin{remark}
$DR_n$ is related to the representation theory of the rational Cherednik  algebra of type A $H_{n,c}$. By results of Berest--Etingof--Ginzburg, if $c = \pm r/n$ with $\gcd(r,n) = 1$, then there exists a unique finite dimensional irreducible representation $L_c$ of $H_{n,c}$. Otherwise there do not exist finite dimensional irreducible representations. Gordon showed that $L_{\frac{n+1}{n}} \otimes {\rm sgn} \cong DR_n$ as $\fS_n$-modules.
\end{remark}

One solution to the problem, given by Haiman, is to use the Hilbert scheme of points in $\bC^2$. He showed that $\cF(DR_n, z; q,t) = \nabla e_n$ where $\nabla$ is defined in terms of Macdonald polynomials.

Another conjectural solution, proposed by Haglund--Haiman--Loehr--Remmel--Ulyanov, gives a combinatorial formula for $\cF(DR_n, z; q,t)$ as follows. For notation, let $\delta_n = (n-1, n-2, \dots, 1,0)$ be the staircase partition and $(1^n) = (1, 1, \dots, 1)$. We draw Young diagrams in French notation and the coordinate of a box is the Cartesian coordinates of its lower-left corner.

\begin{definition}
Fix $\lambda \subset \delta_n$ and $T$ a semistandard Young tableau of $\lambda + (1^n) / \lambda$. For $x = (i,j) \in \bN \times \bN$, define $d(x) = i+j$. A pair $(x,y) \in ((\lambda + (1^n)) / \lambda)^2$ is called a {\bf $d$-inversion for $T$} if (write $x = (i,j)$ and $y = (i',j')$)
\begin{compactitem}
\item $T(x) \subset T(y)$
\item ($d(y) = d(x)$ and $j > j'$) or ($d(y) = d(x)+1$ and $j<j'$)
\end{compactitem}

Set ${\rm dinv}(T)$ to be the number of $d$-inversions for $T$. Also define $z^T = \prod_{x \in \lambda + (1^n) / \lambda} z_{T(x)}$.
\end{definition}

Given $\lambda \subset \delta_n$, we set 
\begin{align*}
D_n^\lambda(z;q) &= \sum_{T \in {\rm SSYT}(\lambda + (1^n) / \lambda} q^{{\rm dinv}(T)} z^T\\
D_n(z;q,t) &= \sum_{\lambda \subset \delta_n} t^{\#(\delta_n / \lambda)} D_n^\lambda(z;q).
\end{align*}

\begin{theorem}[HHLRU] $D^\lambda_n(z;q)$ is symmetric and Schur positive.
\end{theorem}

\begin{conjecture}[HHLRU] $\cF(DR_n, z; q,t) = D_n(z;q,t)$.
\end{conjecture}

\begin{remark} There is a generalization of $D_n(z;q,t)$ corresponding to general $r$ (i.e., for $L_{r/n}$ in the Cherednik picture) which we denote by $D_{(n,r)}(z;q,t)$.
\end{remark}

\begin{conjecture} For $0 \le k \le \min(n,r) - 1$, 
\[
\langle D_{(n,r)}(z;q,t), s_{(k+1,1^{n-k-1})}(z) \rangle = \langle D_{(r,n)}(z;q,t), s_{(k+1,1^{r-k-1})}(z) \rangle.
\]
\end{conjecture}

\begin{example} $D_{(2,3)}(z;q,t) = D_2(z;q,t) = s_2 + (t+q)s_{1,1}$ and $D_{(3,2)}(z;q,t) = s_{2,1} + (t+q) s_{1,1,1}$.
\end{example}

\begin{remark} This conjecture comes from the conjectural relation between $L_{r/n}$ and the Khovanov--Rozansky homology of the $(n,r)$-torus knot.
\end{remark}

\begin{conjecture} $D_{(n,r)}(z;q,t) = D_{(n,r)}(z;t,q)$
\end{conjecture}

\begin{remark} $\cF(DR_n, z;q,t) = \cF(DR_n, z; t,q)$
\end{remark}

\begin{conjecture} $D_{(n,r+n)}(z;q,t) = \nabla D_{(n,r)}(z;q,t)$
\end{conjecture}

\begin{remark} HHLRU conjectured that $\nabla^m e_n(z) = D_{(n,mn+1)}(z;q,t)$.
\end{remark}

\section{Affine Springer fibers of type A.}

Notation: 
\begin{compactitem}
\item $F = \bC(\!(\eps)\!)$, $\cO = \bC[\![\eps]\!]$
\item $G = \SL_n$ and $B \subset G$ is a Borel subgroup. 
\item Let $p \colon G(\cO) \to G(\bC)$ be the reduction modulo $\eps$ and let $I = p^{-1}(B(\bC))$ be the Iwahori subgroup. 
\item $X = G(F) / G(\cO)$ is the affine Grassmannian
\item $\hat{\cB} = G(F) / I$ is the affine flag variety
\item $\fg = {\rm Lie}(G)$ and $\fg = \fh \oplus \bigoplus_{\alpha \in R} \fg_\alpha$ is a root space decomposition. Fix $0 \ne e_\alpha \in \fg_\alpha$. 
\item Let $\alpha_1, \dots, \alpha_{n-1}$ be the simple roots and $\Lambda^\vee$ be the coroot lattice.
\item $\rho^\vee \in \Lambda^\vee \otimes_\bZ \bQ$ such that $\langle \rho^\vee, \alpha_i \rangle = 1$ for $i=1,\dots,n-1$.
Write $r = mn + b$ where $1 \le b \le n-1$ and $\gcd(b,n) = 1$. Define 
\[
v = \eps^m(\eps \sum_{\substack{\alpha \in R\\ \langle \rho^\vee, \alpha \rangle = b-n}} e_\alpha + \sum_{\substack{\alpha \in R\\ \langle \rho^\vee, \alpha \rangle = b}} e_\alpha) \in \fg(F).
\]
\end{compactitem}

We also define
\begin{align*}
X_v &= \{ gG(\cO) \mid {\rm Ad}(g)^{-1}(v) \in \fg(\cO) \} \subset X\\
\hat{\cB}_v &= \{gI \mid {\rm Ad}(g)^{-1}(v) \in {\rm Lie}(I) \} \subset \hat{\cB}.
\end{align*}

If $v$ is regular semisimple, then $X_v$ and $\hat{\cB}_v$ are finite dimensional. If $v$ is elliptic, then $X_v$ and $\hat{\cB}_v$ are compact. There is a natural projection $\pi \colon \hat{\cB}_v \to X_v$. Each fiber of $\pi$ is a classical Springer fiber, so there is an action of $\fS_n$ on $\rH_*^{\rm BM}(\hat{\cB}_v; \bC)$.

There is a stratification $X = \coprod_{\lambda^\vee \in \Lambda^\vee} I \eps^{\lambda^\vee} G(\cO) / G(\cO)$. Set $C_{\lambda^\vee} = X_v \cap (I \eps^{\lambda^\vee} G(\cO) / G(\cO))$. By results of Goresky--Kottwitz--Mac Pherson, if $C_{\lambda^\vee}$ is nonempty, then $C_{\lambda^\vee}$ is an affine space. Its dimension is $\#\{(\alpha, k) \in R \times \bZ \mid 0 < \langle \rho^\vee, \alpha \rangle + kn < r,\ k < \langle \lambda^\vee, \alpha \rangle \}$. So this gives an affine paving for $X_v$, which implies that $\rH_{\rm odd}^{\rm BM}(X_v) = 0$ and $\rH_{\rm odd}^{\rm BM}(\hat{\cB}_v) = 0$.

\begin{remark} The affine cells of $X_v$ are parametrized by Young diagrams contained below the line through $(0,n)$ and $(r,0)$. For $r=n+1$, this is equivalent to $\lambda \subset \delta_n$. Index them by $C_\lambda$.
\end{remark}

\begin{theorem} 
\begin{compactenum}[\rm 1.]
\item $\cF(\rH_{2*}^{\rm BM}(\pi^{-1}(C_\lambda); \bC) \otimes {\rm sgn}, z;q) = q^{(n-1)(r-1)/2} D^\lambda_{(n,r)}(z;q^{-1})$.
\item There exists a filtration on $\rH_*^{\rm BM}(\hat{\cB}_v; \bC)$ such that 
\[
\cF({\rm gr}_* \rH_{2*}^{\rm BM}(\hat{\cB}_v; \bC) \otimes {\rm sgn}, z; q,t) = q^{(n-1)(r-1)/2} D^\lambda_{(n,r)}(z;q^{-1},t).
\]
\end{compactenum}
\end{theorem}

\begin{remark} This theorem gives another proof for symmetricity and Schur positivity of $D^\lambda_n(z;q)$. Sommers proved part 2 when $q=t=1$.
\end{remark}

\begin{example}
Set $n=3$ and $r=4$. 
\[
\begin{array}{l|lllll}
\delta_3 \supset \lambda & \emptyset & (1) & (1,1) & (2) & (2,1) \\
\Lambda^\vee \ni \lambda^\vee & \alpha_1^\vee + \alpha_2^\vee & - \alpha_2^\vee & -\alpha_1^\vee & -\alpha_1^\vee - \alpha_2^\vee & 0\\
\dim & 3 & 2 & 2 & 1 & 0
\end{array} \qedhere
\]
\end{example}

\end{document}