All Questions
14 questions with no upvoted or accepted answers
9
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233
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A detail in I. Piatetski-Shapiro and S. Rallis's "Doubling paper": computing the integral on negligible orbits
I'm currently reading the paper "L-functions for the classical groups" by I. Piatetski-Shapiro and S. Rallis, where they introduced the doubling method over classical groups.
I'm confused at ...
- 639
5
votes
0
answers
185
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Finite coefficients Langlands for function fields
Do we have a finite coefficients Langlands correspondence for function fields? By which I mean a bijection between Galois representations $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...
- 4,408
5
votes
0
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213
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Truncation and weighted orbital integrals in hyperbolic term of trace formula for $\mathrm{GL}(2)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I am looking at Gelbart--Jacquet's article in the first Corvallis volume (the article entitled Forms of $\GL(2)$ from an analytic point of ...
- 241
5
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0
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132
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Field of definition of compatible system of Galois representations
Let $K,F$ be number fields and suppose that there is a compatible system of Galois representations
$$(\rho_{\lambda} : \text{Gal}(\overline{K}/K) \longrightarrow \text{GL}_n(\overline{F}_\lambda) )$$
...
- 173
4
votes
0
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219
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A question of integral on $p$-adic fields $\mathbb{Q_p}$
We assume that $(\pi,V)$ is an admissible, irreducible and infinite-dimensional representation of $GL_2(\mathbb{Q_p})$. In the proof of existence and uniqueness of Kirillov model, the key step is that ...
- 421
4
votes
0
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136
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Classification of coregular spaces
Let $G$ be a reductive group, and let $V$ be a finite dimensional vector space on which $G$ acts linearly. Let $\mathcal{P}$ be the ring of polynomial invariants of the action of $G$ on $V$. We say ...
- 26.9k
3
votes
0
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139
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Cartan decomposition for $G[z]$
Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\...
- 31
3
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0
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199
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Decompositions of representations of pro-p groups
Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...
- 93
2
votes
0
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168
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Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?
In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
- 6,180
2
votes
0
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138
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Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character
Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial
$$
P(\rho|_F,T) = \det{(1 - \operatorname{...
- 103
1
vote
0
answers
124
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A question related to Kirillov model
I am reading Jackson - The theory of admissible representations of $\operatorname{GL}(2, F)$ and am not able to understand the following map related to Kirillov model. This result appears on p. 54:
I ...
- 424
1
vote
0
answers
115
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Powers of automorphic Eisenstein series
Let $G$ be a reductive group defined over $\mathbb{Q}$. Let $P$ be a standard parabolic subgroup of $G$ with Levi decomposition
$$P = MN.$$
We denote by $R_{disc,M}$ the discrete spectrum of $M$. Let $...
- 103
1
vote
0
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105
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Local factors determine Weil representations - proof of the Artin representation case
This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
- 103
0
votes
0
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91
views
Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739