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On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$

Teppei Takamatsu

TL;DR

This work extends Chan–Ivanov's link between semi-infinite and affine Deligne–Lusztig varieties from $\text{GL}_n$ to the symplectic group $\text{GSp}_{2n}$ and its inner forms. It establishes an inverse-limit description of a semi-infinite variety in terms of higher-level affine DL varieties at infinite level, parameterized by a symplectic counterpart $V^{\mathrm{symp}}_b$ with equivalence relations $\sim_{b,m,r}$ and $\dot{\sim}_{b,m,r}$. A sharp local description of connected components is provided: certain components $X^{0}_{w_r}(b)_{\mathcal{L}}$ (after perfection) decompose as a product of a classical Deligne–Lusztig variety with an affine space, and the finite-type families $X_h$ connect these geometric objects back to Lusztig’s conjectural framework for constructing representations relevant to the local Langlands correspondence. The results reinterpret prior representation-constructing frameworks (notably $X_h$) as instances of Lusztig's broader conjectural picture and unify the geometry of DL varieties across finite, semi-infinite, and infinite-level limits in the $\text{GSp}$ setting.

Abstract

We prove that Lusztig's semi-infinite Deligne--Lusztig variety for $\mathrm{GSp}$ (and its inner form) is isomorphic, as a set with action, to an affine Deligne--Lusztig variety at infinite level, generalizing a result of Chan--Ivanov. Furthermore, we show that a component of some affine Deligne--Lusztig variety $X^0_{w_r}(b)_{\mathcal{L}}$ for $\mathrm{GSp}$ can be written, up to perfection, as a direct product of a classical Deligne--Lusztig variety with an affine space. We also study the varieties $X_h$ defined by Chan and Ivanov, and show that $X_h$ at infinite level can be realized as a subset of semi-infinite Deligne--Lusztig varieties defined using components of affine Deligne--Lusztig varieties such as $X^0_{w_r}(b)_{\mathcal{L}}$ above, even in the $\mathrm{GSp}$ case. This reinterprets previous constructions of representations from $X_h$ as instances of Lusztig's conjectural picture.

On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$

TL;DR

This work extends Chan–Ivanov's link between semi-infinite and affine Deligne–Lusztig varieties from to the symplectic group and its inner forms. It establishes an inverse-limit description of a semi-infinite variety in terms of higher-level affine DL varieties at infinite level, parameterized by a symplectic counterpart with equivalence relations and . A sharp local description of connected components is provided: certain components (after perfection) decompose as a product of a classical Deligne–Lusztig variety with an affine space, and the finite-type families connect these geometric objects back to Lusztig’s conjectural framework for constructing representations relevant to the local Langlands correspondence. The results reinterpret prior representation-constructing frameworks (notably ) as instances of Lusztig's broader conjectural picture and unify the geometry of DL varieties across finite, semi-infinite, and infinite-level limits in the setting.

Abstract

We prove that Lusztig's semi-infinite Deligne--Lusztig variety for (and its inner form) is isomorphic, as a set with action, to an affine Deligne--Lusztig variety at infinite level, generalizing a result of Chan--Ivanov. Furthermore, we show that a component of some affine Deligne--Lusztig variety for can be written, up to perfection, as a direct product of a classical Deligne--Lusztig variety with an affine space. We also study the varieties defined by Chan and Ivanov, and show that at infinite level can be realized as a subset of semi-infinite Deligne--Lusztig varieties defined using components of affine Deligne--Lusztig varieties such as above, even in the case. This reinterprets previous constructions of representations from as instances of Lusztig's conjectural picture.
Paper Structure (16 sections, 22 theorems, 286 equations, 3 figures)

This paper contains 16 sections, 22 theorems, 286 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Statement of the comparison theorem
  3. Basic Notation
  4. Definition of semi-infinite Deligne--Lusztig varieties and affine Deligne--Lusztig varieties
  5. Comparison theorem
  6. Proof of comparison theorem
  7. Proof for the semi-infinite case
  8. Proof for the affine case
  9. The relation $\sim_{b,m,r}$ and $\dot{\sim}_{b,m,r}$
  10. description of the connected components
  11. Representatives of $b$
  12. Linear algebraic description of the connected components
  13. Structure of $\mathcal{L}_{b}^{\mathrm{symp}}$, $n:$ even case
  14. Structure of $\mathcal{L}_{b}^{\mathrm{symp}}$, $n:$ odd case
  15. Structure of affine Deligne--Lusztig varieties
  16. ...and 1 more sections

Key Result

Theorem 1.0.1

If $r+k\geq m+1$, there are the following two $J_b(K)$-equivariant commutative diagrams with bijective horizontal arrows. \xymatrix{ \{v\in V^{\mathrm{symp}}_{b}\mid \alpha:=\langle v,F^{n}(v)\rangle \in K^{\times}\} \ar[r]^-{g_{b,0}}_-{\sim} \ar[d] & X_{w}^{(U)}(b) \ar[d]\\ V^{\mathrm{symp}}_{b}/\b

Figures (3)

  • Figure : The shape of $G_{x,0}$ when $k=1$, for $2n=4, 6$
  • Figure : The shape of the right-hand side of (\ref{['eq:Hcontain']}) for $\mathrm{GSp}_{8}$ The $(i, \tau (i))$-th entries for $1 \leq i \leq 8$ are enclosed in a square. The exceptional entry is marked with a wave line.
  • Figure : The shape of $(\prod_{\alpha_{i,j} \in I} L_{[g(i,j), h(i,j))} U_{\alpha_{i,j}})$ for $\mathrm{GSp}_{8}$.

Theorems & Definitions (70)

  • Theorem 1.0.1: See Theorem \ref{['theorem:comparison']} for the precise statements
  • Theorem 1.0.2: see Theorem \ref{['theorem:restradlv']}
  • Theorem 1.0.3: see Remark \ref{['remark:Xhinf']}
  • Lemma 2.3.1
  • proof
  • Definition 2.3.2
  • Remark 2.3.3
  • Lemma 2.3.4
  • proof
  • Remark 2.3.5
  • ...and 60 more