All Questions
14 questions
51
votes
2
answers
4k
views
Which philosophy for reductive groups?
I am just beginning to look further into trace formulas and automorphic forms in a quite general setting. For long I have noticed that the natural assumption on the group $G$ we work on is to be ...
- 5,424
27
votes
1
answer
3k
views
Reconciling Lusztig's results with the Langlands philosophy
Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}_q$, $G = \boldsymbol{G}(\mathbb{F}_q)$, $W = \mathrm{W}(\mathbb{F}_q)$ the Witt vectors over $\mathbb{F}_q$, and $K = \mathrm{...
- 805
18
votes
6
answers
2k
views
Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex ...
- 41.4k
7
votes
1
answer
165
views
Which groups can have $GSp(4)$ as local component?
In some cases the relations between a global group $G$ (over the adeles $\mathbb{A}$ of a field $F$) and its local components $G_v$ (where $v$ are the places of $F$) are well known. Obviously a group ...
- 5,424
5
votes
1
answer
222
views
Do we have $G(\mathbb A_S) G(k) = G(\mathbb A)$ for sufficiently large $S$?
Let $G$ be a linear algebraic group over a number field $k$. If necessary, assume $G$ is connected and reductive. Let $\mathbb A$ be the ring of adeles of $k$, and $\mathbb A_S = \prod\limits_{v \in ...
- 6,180
5
votes
1
answer
372
views
Diagonalizable pro-algebraic group in Kottwitz's 1985 Compositio paper
In Kottwitz's 1985 Compositio paper,
Isocrystals with additional structure, first page, paragraph 4:
Let $\mathbb{D}$ be the diagonalizable pro-algebraic group over $\mathbb{Q}_p$ with character ...
- 53
4
votes
0
answers
313
views
How to determine the unramified character corresponding to an unramified Langlands parameter?
Let $F$ be a p-adic field with ring of integers $\mathcal{O}$. Let $\textbf{G}$ be a connected split reductive algebraic group over $F$. For simplicity, we assume that $\textbf{G}$ is a Chevalley ...
- 960
3
votes
2
answers
471
views
Representations of complex semi-simple algebraic group "defined over $\mathbf{Z}$"?
If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over $\mathrm{Spec}(\...
- 163
3
votes
1
answer
297
views
Orbit of a parahoric subgroup on a flag variety
Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$).
Given a parahoric subgroup $K \subset G(F)$, and a ...
- 37k
3
votes
0
answers
103
views
compactly induction of smooth modules over Hecke algebras
Let $G$ be a locally profinite group and $H$ its closed subgroup. It is well-known that a smooth representation of $G$ is identified with a smooth module over $\mathcal{H}(G)$, where $\mathcal{H}(G)$ ...
- 479
3
votes
0
answers
227
views
Kottwitz's vertical map
I'm looking at the map $w_H$ defined by Kottwitz in "Isocrystals with additional structure II" in section 7. It is a surjective group homomorphism defined for all reductive groups $H$ from $H$ to $X^{*...
- 2,429
2
votes
1
answer
148
views
Does the F-unitary group isomorphism arises from a conformal isometry?
Let $K$ be a CM-field with totally real subfield $F$. Let $(V_1, h_1)$ and $(V_2, h_2)$ be two $n$-dimensional $K$-vector spaces with nondegenerate Hermitian forms, where $n\geq 3$.
Question 1 Does ...
- 337
2
votes
1
answer
92
views
Image of the intertwining operator for GL(2) is $K$-invariant at the "pole" $s=1$
I am taking a look at the residues of Eisenstein series and have a question about a local computation. Let $k$ be a local field, $G = \operatorname{GL}_2(k)$, and $P$ (resp. $K$) the standard ...
- 6,180
2
votes
0
answers
81
views
Continuity of the conductor of automorphic representations
I am interested in properties of (semi-)continuity of the conductor of automorphic representation. Let $F$ be a number field.
The meta question is, given a function on the unitary dual of $PGL(2, F)$ ...
- 5,424