[Paper Review] On Some Stratifications of Affine Deligne-Lusztig Varieties for SL_3

Boris Zbarsky|ArXiv.org|May 31, 2009

Coding theory and cryptography2 references7 citations

TL;DR

This paper investigates the action of the Iwahori subgroup on the Borel-Moore homology of affine Deligne-Lusztig varieties for $SL_3$, showing that the subgroup $A(\mathfrak{o}_F)$ acts trivially on homology by constructing a stratification preserved under $A(\mathfrak{o}_F)$ and extending the action to $A(\mathfrak{o}_L)$, which is connected. The key result is that the representation of $A(F) \cong \mathbb{Z}^2$ on homology factors through permutation of strata, with trivial action from $A(\mathfrak{o}_F)$.

ABSTRACT

Let L be k((ε)), where k is an algebraic closure of a finite field with q elements and εis an indeterminate, and let σbe the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of σin L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let W be the extended affine Weyl group of G. Given x in W and b in G(L), we have some subgroup of G(L) that acts on the affine Deligne-Lusztig variety X_x(b) = {gI in G(L)/I : g^{-1}bσ(g) is in IxI} and hence a representation of this subgroup on the Borel-Moore homology of the variety. We investigate this representation for certain b in the cases when G is SL_2 and G is SL_3.

Motivation & Objective

• To understand the representation of the group $A(F) \cong \mathbb{Z}^{n-1} \times A(\mathfrak{o}_F)$ on the Borel-Moore homology of affine Deligne-Lusztig varieties $X_x(\epsilon^\nu)$ for $G=SL_3$.
• To determine when the subgroup $A(\mathfrak{o}_F)$ acts trivially on this homology, especially in cases where the larger group $A(\mathfrak{o}_L)$ does not act by left multiplication.
• To develop a stratification of $X_x(\epsilon^\nu)$ into disjoint closed subsets preserved by $A(\mathfrak{o}_F)$, enabling the extension of the action to $A(\mathfrak{o}_L)$.
• To prove that the homology representation factors through $\mathbb{Z}^{n-1}$, with the action realized as permutation of homology spaces of strata.
• To generalize techniques from the $SL_2$ case, where $A(\mathfrak{o}_L)$-action implies triviality, to $SL_3$ via stratification.

Proposed method

• Developed criteria using valuations of minors (including $1\times1$ minors) to determine membership in Iwahori double cosets $IxI$.
• Introduced a method to compute the $x \in \widetilde{W}$ such that $g \in w^{-1}U_1 w x I$, based on matrix entries and valuation conditions.
• Constructed a stratification of $X_x(\epsilon^\nu)$ into disjoint closed subsets preserved under $A(\mathfrak{o}_F)$-action.
• Extended the $A(\mathfrak{o}_F)$-action on each stratum to an action of the connected group $A(\mathfrak{o}_L)$, implying trivial homology action.
• Used Theorem 6 to verify that the variety $X_x(\epsilon^\nu) \cap U_1 w I$ is nonempty only for $w=1$ when $x=\epsilon^{(d,e,f)}$ with $f \leq e \leq d$, and $i>j>k$, $i+j+k=0$.
• Applied valuation conditions from Theorem 1 to determine necessary and sufficient conditions for $h \in IxI$, involving $\mathop{\mathrm{val}}(\alpha\epsilon^k) > f+e$, $\mathop{\mathrm{val}}(\gamma) > f$, and $\mathop{\mathrm{val}}(\beta) > f$.

Experimental results

Research questions

• RQ1 Under what conditions does the subgroup $A(\mathfrak{o}_F)$ act trivially on the Borel-Moore homology of $X_x(\epsilon^\nu)$ for $SL_3$?
• RQ2 Can the action of $A(\mathfrak{o}_F)$ on $X_x(\epsilon^\nu)$ be extended to an action of $A(\mathfrak{o}_L)$ via stratification, even when $A(\mathfrak{o}_L)$ does not act by left multiplication?
• RQ3 For which $x \in \widetilde{W}$ and $\nu \in \mathbb{Z}^3$ with $i>j>k$, $i+j+k=0$, is $X_x(\epsilon^\nu) \cap U_1 w I$ nonempty only for $w=1$?
• RQ4 How do valuation conditions on matrix minors determine membership in $IxI$ for $SL_3$?
• RQ5 What is the structure of the homology representation of $A(F) \cong \mathbb{Z}^2$ on $X_x(\epsilon^\nu)$, and how does it factor through $\mathbb{Z}^2$?

Key findings

• The subgroup $A(\mathfrak{o}_F)$ acts trivially on the Borel-Moore homology of $X_x(\epsilon^\nu)$ for $SL_3$, as the action extends to the connected group $A(\mathfrak{o}_L)$.
• For $x = \epsilon^{(d,e,f)}$ with $f \leq e \leq d$ and $i>j>k$, $i+j+k=0$, the intersection $X_x(\epsilon^\nu) \cap U_1 w I$ is nonempty only when $w=1$.
• The action of $A(F) \cong \mathbb{Z}^2$ on homology factors through permutation of homology spaces of disjoint closed strata, with trivial contribution from $A(\mathfrak{o}_F)$.
• The necessary and sufficient conditions for $h \in IxI$ include $\mathop{\mathrm{val}}(\alpha\epsilon^k) > f+e$, $\mathop{\mathrm{val}}(\gamma) > f$, and $\mathop{\mathrm{val}}(\beta) > f$, derived from Theorem 1.
• The hexagon associated with $g = \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$ has all vertices coinciding, confirming the applicability of Theorem 6.
• In the case $f=k$, $w=1$, and $d=j$, $e=i$, the conditions $\mathop{\mathrm{val}}(a) > 0$, $\mathop{\mathrm{val}}(c) > 0$, and $\mathop{\mathrm{val}}(b) > 0$ are required for $h \in IxI$, and allow elimination of $a,b,c$ via right $I$-action.