30
votes
Accepted
What's the point of a Whittaker model?
This question is a bit like saying "what's the point of the theory of bases for vector spaces -- this just gives you an isomorphism of your space with $\mathbb{R}^n$. What is the point of defining ...
- 41.4k
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What is the defining property of reductive groups and why are they important?
A linear algebraic group is unipotent if it consists entirely of unipotent linear transformations, i.e. $I+N$ with $N$ nilpotent.
A linear algebraic group is reductive if it has no connected normal ...
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18
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Which philosophy for reductive groups?
Here is a natural interpretation of complex reductive groups:
A complex algebraic group is reductive if and only if it is the complexification of a compact Lie group.
More precisely, every compact ...
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17
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What is the defining property of reductive groups and why are they important?
For algebraic groups over $\mathbb{C}$, we have the following deep theorem of Cartan, Chevalley and Mostow describing concretely the reductive groups:
For an algebraic subgroup $G$ of $\mathrm{GL}_n(\...
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16
votes
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Why is Mumford's GIT-quotient so effective?
By my understanding, your question is not "Why is Mumford's construction better than the affine quotient". As you note, Proj is better than Spec of invariants for taking quotients by $\mathbb{G}_m$. ...
- 148k
16
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Accepted
Why are coroots needed for the classification of reductive groups?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Zent{Z}\newcommand\Q{\mathbb Q}\newcommand\Z{\mathbb Z}$The collections of roots and the coroots, as abstract root systems, provide the same ...
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15
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Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
The right person to answer this question is probably Dinakar Ramakrishnan (a good working reference is his paper "Modularity of the Rankin–Selberg $L$-series, and multiplicity one for $\mathrm{SL}...
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14
votes
Accepted
Split rank of inner forms
Every torus in $G$ transfers to $G^*$, so we definitely have the desired inequality. If we have equality, then there is a maximal split torus $A$ in $G$ that is also maximal split when transferred to ...
- 12.9k
13
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A reductive group has a quasi-split inner form
Nothing is "better-suited to using the classical language"; if you cannot express things clearly via schemes then think harder about it until you can. Also, any connected reductive group over a field ...
13
votes
Why are coroots needed for the classification of reductive groups?
(1) As anon says, an example is $G_1 = \mathrm{GL}_2$ and $G_2 = \mathbb{G}_m \times \mathrm{PGL}_2$. We can identify the root lattice and co-root lattice with $\mathbb{Z}^2$ (with the pairing being ...
- 156k
13
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Definition of locally symmetric space of reductive groups
There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a theorem, not a ...
- 37k
12
votes
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Endoscopic group that is not a subgroup
If I understand the definition correctly, a connected reductive group $H$ is an endoscopic group for a connected reductive group $G$ if its Langlands dual $H^\vee$ is a connected centralizer in $G^\...
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11
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What's the point of a Whittaker model?
In addition to the other good answer, in the context of automorphic forms the uniqueness of local Whittaker models is specifically useful in computations of global integrals: when/if the integral can ...
- 23k
11
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Simple Proof that a Reductive Group is Unimodular?
The modular character $\Delta: G(k)\rightarrow {\mathbb R}_{>0}$ on $G(k)$ is trivial on the commutator subgroup and is trivial on a compact open subgroup and also on the centre. These three groups ...
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11
votes
Accepted
Cusp forms have an orthonormal basis of eigenfunctions for all Hecke operators
These things are not trivial at all, but by the time Langlands was writing "Euler products" they were known, and quite familiar to many people at Princeton and Yale, even if not so many other places.
...
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10
votes
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On existence of a certain irreducible character of $SL(5, q)$
[The author or this question made me aware of this thread, so I send the answer here.]
The description in the question is almost correct, except when it comes to the centralizer of the semisimple ...
- 391
10
votes
Accepted
Type of place versus type of unitary group
Things are perhaps a bit messier than you hope. In particular it is not true that the unitary group is non-quasi-split if and only if $v$ ramifies. Disclaimer: I did not know the answer to this ...
- 41.4k
10
votes
Accepted
Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
This is an elaboration on what is in the other answers. First, general categorical arguments can be used to prove the following. Let $K$ be a field and $A$ a $K$-algebra. The profinite completion $\...
- 118k
9
votes
Accepted
Why are spherical representations subquotients of unramified principal series?
Statement 2. comes from the following classical fact whose proof can be found in e.g. Bushnell and Kutzko, "The admissible dual of ${\rm GL}(N)$ via compact open subgroups". This is a particular case ...
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9
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Primer on Eisenstein series
Look at J. Bernstein, Erez Lapid - "On the meromorphic continuation of Eisenstein series", Journal of the American Mathematical Society (2023) Volume 37, Number 1, January 2024, Pages 187-...
9
votes
Accepted
Generic supercuspidal representations of $\operatorname{GL}_n$ can be defined by integrals over $U$
The answer to both questions is yes.
All irreducible supercuspidal representations of ${\rm GL}(N,F)$ are generic. See e.g. I. M. Gelfand and D. A. Kajdan,
Representations of the group ${\rm GL}(n,...
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votes
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Are all cuspidals induced?
Question 1. Yes indeed.
a) There are new results for classical groups and their inner forms (works of Shaun Stevens, Daniel Skodlerack, ...). In particular Skodlerack proved that in the case of "...
- 6,349
9
votes
Accepted
An example of a Deligne–Lusztig variety for a general linear group
Let $\mathcal{F}\colon V_1\subseteq\dotsb\subseteq V_n$ and $\mathcal{F}'\colon U_1\subseteq\dotsb\subseteq U_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative ...
- 1,666
9
votes
Accepted
Parabolic subgroups of reductive group as stabilizers of flags
$\DeclareMathOperator\GL{GL}$Yes for reductive $G$, and this follows quickly from the dynamical characterization of parabolics: A subgroup $H$ of a reductive group $G$ is parabolic if and only if ...
- 148k
9
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Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
For a reductive group the category of representations is semisimple so the algebra of endomorphisms of the fiber functor is just a product of matrix algebras, one for each irreducible representation.
...
- 148k
8
votes
Accepted
$GSp(4)$ vs $PSp(4)$
If we think about classical modular forms, one typically works on SL(2). One could also work on PSL(2), but one would like to write down congruence subgroups in terms of matrices, so one often ...
- 6,039
8
votes
Accepted
Automorphic quotients for inner forms or $GSp(4)$
Let me give you the inner forms of $\mathrm{GSp}(4)$ with compact adelic quotient.
Every nonsplit inner form of $\mathrm{GSp}(4)$ is obtained in the following way: take $D$ a division quaternion ...
- 5,382
8
votes
What is the archimedean Hecke algebra?
Here is the picture, as I understand it, for $\mathrm{GL}_n$; this is described in chapter 8 of Godement-Jacquet (and see also the archimedean theory in Jacquet-Langlands).
Let $F \in \{\mathbb{R},\...
- 8,374
8
votes
What is the archimedean Hecke algebra?
The terminology is a bit misleading, and the analogy with the non-archimedean situation is a bit forced.
The goal was/is to have a $\mathfrak g,K$-module be a "Hecke algebra module", for some ...
- 23k
8
votes
Accepted
Schubert cells in G/P for reductive G
You already answered your question: the center of any reductive group lies in any parabolic, so if $G$ is reductive, and $G_{\operatorname{ad}}$ its adjoint quotient (which is, of course, semi-simple),...
- 44.7k
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