User - MathOverflow

linearization method in smooth transfer of relative trace formula

2013-06-12T12:44:03Z

I'm currently reading Jacquet's paper 'smooth transfer of Kloosterman integrals', in which Jacquet proved there is a smooth transfer for compactly supported smooth functions on GL(n,F) to smooth functions on Hermitian matrices for E/F, where F is a p-adic field and E/F is a quadratic extension.

The method is linearization, which is suggested by J.Waldspurger, that is, to prove the existence of such a transfer for smooth functions on $M(n\times n,F)$, the Lie algebra of GL(n,F). It seems that this method is a useful tool (maybe most powerful?) for similar problems.

But I'm not quite understand why smooth transfer on Lie algebra level implies that on group level, and this is not explained in Jacquet's paper. Maybe it is not too difficult, but I couldn't figure it out. I appreciate it a lot if someone could explain it in some details or suggest some references on it.

reference request about fact the character of irreducible representation determine the representation itself.

2013-03-27T03:38:24Z

It is well known that if two (irreducible) admissible representations have the same characters, then they are isomorphic. To my knowledge, this is true for both Lie groups and p-adic groups.

In the case of finite groups, this follows from the orthogonality relations of characters. I would like to know what are the main ingredients in the proof of its generalization in infinite dimensional case. In particular, is it simple if we assume both representations to be irreducible.

I checked the books by Knapp, Wallach, but couldn't find any hint(I'm not an expert in this field, and not familiar with these books). I appreciate a lot for any explanations of the proof, or suggestions on references containing it.

a question about the proof of analytic continuation of Eisenstein series for GL(2)

2012-12-26T13:01:33Z

I'm reading Gelbart and Jacquet's paper 'forms of GL(2) from the analytic point of view', and was confused at a point in the proof of analytic continuation of Eisenstein series. On the top of page 232, after the calculation of L^2 norm for the derivative of the truncated Eisenstein series, it said it is easy to conclude the the convergence of the Taylor series of truncated series in some disc, which I don't get it. It seems to me that we can conclude the convergence in L^2 norm, but to get pointwise convergence, we need more work.

A similar argument appears also in Kubota's book 'Elementary theory of Eisenstein series', which is at the bottom of page 31.

Could someone explain the hiding part? Thanks a lot.

Gelbart and Jacquet's paper can be found via books.google.com

About Theorem 3.2 in 'introduction to spectral theory of automorphic forms' by Iwaniec

2012-10-13T14:21:10Z

In Theorem 3.2 of 'Introduction to spectral theory of automorphic forms' by Iwaniec,the first bound is about the coefficients of automorphic forms

$$\sum_{|n|\le N}|n||c_n|^2<<(N+|s|)e^{\pi|s|}$$

whereas in the paper 'On the uniform equidistribution of long closed horocycles', by A. Strombergsson, it says that this bound needs some revision (the note after Proposition 4.5 in the paper.) Also another bound is given:

$$\sum_{|n|\le N}|c_n|^2=O((N+|s|)e^{\pi|s|})$$

The reason is that the formula on page 61, line 7, in Iwaniec's book, is not correct.

I'm not familiar with these stuff, and want to know which statement is correct, Iwaniec or Strombergsson? Or both? Also what's the best estimate towards these coefficients (maybe in various forms)?

Thanks a lot for your help, I appreciate it a lot!

reference help on a result of Whittaker functions of supercuspidal representations

2011-11-13T04:43:07Z

Let $F$ be a p-adic field, $\pi$ an irreducible supercuspidal representation of $GL(n,F)$, then it admits a unique Whittaker model $\mathcal{W}(\pi)$. For any $W\in \mathcal{W}(\pi)$, a basic result is that $W(g)$ is a compactly supported function mod $NZ$, where $N$ is the maximal unipotent subgroup and $Z$ is the center of $GL(n,F)$.

Does anyone know a reference containing this result? Many thanks.

Is the space of global Whittaker functions complete?

2011-08-04T16:02:44Z

Let $f$ be a complex valued function of $GL_n(\mathbb{A})$, where $\mathbb{A}$ is the adeles of some number field. Assume $f(ug)=\psi(u)f(g)$ for any $u$ in the standard maximal unipotent subgroup $N_n(\mathbb{A})$. If the integral $\int_{N_m(\mathbb{A})\backslash GL_m(\mathbb{A})}f(h)W_\phi(h) \; d h=0$ for any automorphic form $\phi$ on $GL_m$ and $W_\phi$ is the corresponding Whittaker function (here we embed $GL_m$ into $GL_n$ on the left upper corner), can we say $f$ is identically zero?

It is known that the corresponding local statement is true.

A convergence problem about integral operator in the space of representations

2012-01-15T15:07:38Z

This would be a basic problem in representation theory.

Let $G$ be a unimodular real Lie group, $(\pi,V)$ a smooth representation of $G$ in a Frechet space $V$. Let $f$ be a smooth function on $G$. Now define the operator $\pi(f)$ as $$\pi(f)v=\int_G f(g)\pi(g)vdg$$ for any $v\in V$.

Now the question is that in order to show $\pi(f)$ is an operator on $V$, i.e. $\pi(f)v\in V$ for any $v$, it suffices to check $$|\pi(f)v|_{\mu}<\infty$$

for some particular seminorm $||_{\mu}$, or to check that is finite for ALL seminorms on $V$ ?

In particular, if $(\pi,V)$ is the smooth vectors in a unitary representation, then for all smooth function $f$, which is also in $L^1(G)$, $\pi(f)$ is an operator on $V$, and in fact continuous,right?

Many thanks.

a question about asymptotics of solutions to a ordinary differential equation

2012-01-09T22:22:00Z

This might be an easy question in the theory of ordinary differential equation. But since I know very little about it, I posted it here and hope to get some answers or references.

Consider the equation $$x\phi'(x)-c\phi(x)=xf(x), x>0$$ where $c$ is some constant.

My question is that under what (growth) conditions, the solution $\phi(x)$ has the following property: the limit $$\lim_{x\to 0}x^{-c}\phi(x)$$ exists and nonzero?

Thank you very much for your help and sorry if this question is not appropriate here.

embedding of local representation into automorphic representation

2011-08-08T13:49:13Z

Assume $v$ is a place of a number field $k$, finite or not. Let $\pi_v$ be an irreducible admissible generic representation of $GL_n(k_v)$. Is it always true that we can find some irreducible generic automorphic representation $\Pi$ of $GL_n(\mathbb{A}_k)$ with $v$-component exactly isomorphic to $\pi_v$?

A form of the famous generalized Ramanujan conjecture says that if $\Pi$ is cuspidal, then every component is tempered. So the above question is kind of converse to Ramanujan conjecture.

It is known that if $v$ is a finite place, and $\pi_v$ is supercuspidal, then $\Pi$ always exists, and in fact we can take $\Pi$ to be a cuspidal representation.

Many thanks for any answer or references related to this question.

A trilinear forms question about representations of real linear group.

2011-11-16T20:31:01Z

This might be a naive question. Suppose $\pi_i$ are irreducible generic unitary representation of $GL_{n}(\mathbb{R})$ with its associated Whittaker models $\mathcal{W}(\pi_i)$. Let $E_m$ be the space of homogeneous polynomials of degree $m$ on $\mathbb{R}^n$, then $GL_n(\mathbb{R})$ acts on this space via $(g.P)(x)=P(x.g)$, where $x.g$ denotes $g$ acts on $x$ from the right. Let $e=(0,...,0,1)\in \mathbb{R}^n$. Finally let $s_0$ be some complex number. Let $Z$ be the center of $GL_n$, $N$ the standard maximal unipotent subgroup.

Now suppose that there is a nontrivial continuous trilinear form $L:V_1\times V_2 \times E_m\rightarrow \mathbb{C}$ satisfying $L(g.W_1,g.W_2,g.P)=|detg|^{-s_0}L(W_1,W_2,P)$ for all $W_1,W_2,P$ . Then is it always true that $$\int_{ZN\GL_n}W_1(g)W_2(g)P(eg)|detg|^{s_0}dg < \infty ?$$

In other words, the above integral gives one realization of $L$.

when can we lift an action of Lie algebra?

2011-08-24T13:36:25Z

Suppose $G$ is a Lie group, $\mathfrak{g}$ its Lie algebra, if we have a smooth representation $(\pi,V)$, then it induces an action of $\mathfrak{g}$ on $V$. Now conversely, if we have a nice (with properties you may assume) action of $\mathfrak{g}$ on $V$, can we say such action arises from some unique smooth action of $G$?

Here we may assume $G$ to be simply connected if needed. Thank you.

A question about zeros of Tate type integral

2011-09-05T12:51:58Z

Fix a positive integer $n$. Fix a continuous character $\chi$ of $\mathbb{R}^*$ with the form $\chi(x)=sign(x)|x|^t$ for some complex number $t$. If $\phi$ is a Schwartz function on $\mathbb{R}$, let $s$ be a complex variable. Consider the following integral $$F(s)=\int_{\mathbb{R}^*}\chi(x)|x|^{ns}\phi(x)\frac{dx}{x}$$

This is a Tate-type integral at real place. $F(s)$ converges in some right half complex plane and has a meromorphic continuation to the whole plane. The poles of $F(s)$ are responsible for the poles of the $L$ factor of some character.

But here I'm concerning the zeros of $F(s)$ at some left half plane. My question is: may I find some Schwartz function $\phi$ so that $F(s)$ is not identically zero, but has infinitely many zeros of the form $s_0, s_0-\frac{2}{n}, s_0-\frac{4}{n}, s_0-\frac{6}{n}....$?

Thank you.

a question about Kirillov model of unitary representations over GL_n(R)

2011-09-29T19:13:42Z

Let $F$ be either a p-adic field or real number field. $GL_{n-1}(F)$ embeds into $GL_n(F)$ on the left upper corner, let $P_n(F)$ be the mirabolic subgroup of $G_n(F)$ consisting of matrices with the last low like $(0,0,...,0,1)$. So we have inclusions $GL_{n-1}(F)\subset P_n(F)$.

When $F$ is p-adic, let $(\pi,V)$ be a generic irreducible admissible smooth representation of $GL_n(F)$, with its Whittaker model $\mathcal{W}(\pi)$. It is well known that for any $W_v(g)\in \mathcal{W}(\pi)$, the restriction map sending $W_v$, as a function of $GL_n$, to $W_v|_{GL(n-1)}$ ,

is injective, and from here we obtain the Kirillov model of $\pi$ as a space of functions on $GL_{n-1}$ .

My question is that when $F=\mathbb{R}$, and $\pi$ is unitary, in which case we also have Kirillov model for $\pi$, but I'm not sure if it comes in the same way as we described above in p-adic case. I don't know the exact reference containing the proof that $\pi$ has Kirillov model, and thus have no idea how it comes.

Another question is that when $F=\mathbb{R}$, $\pi$ a generic smooth irreducible admissible, is it still true for any Whittaker function $W_v$ of $\pi$, the restriction of $W_v$ to $GL(n-1)$ as above is injective, which implies we again have a Kirillov model for $\pi$.

Any answer, comment and reference are appreciated.

what's the contragredient of induced representation

2011-09-20T21:03:37Z

Let $G$ be a real reductive Lie group, $P=MN$ its parabolic subgroup with Levi decomposition. Suppose $\sigma$ is a smooth admissible irreducible representation of $M$, extend this to $P$ by letting $N$ act trivially. Form the unitarily induced representation $Ind_P^G(\sigma)$.

My question is what is the contragredient representation (smooth admissible dual) of $Ind_P^G(\sigma)$ in terms of $\sigma$ ? In particular, is it equal to $Ind_P^G(\sigma')$, where $\sigma'$ is the smooth admissible dual of $\sigma$?

reference help needed on a fact about decomposition of representations

2011-09-01T19:29:03Z

For simplicity, let $G_n$ be $GL_n(\mathbb{R})$, $\mathfrak{g}_n$ be its Lie algebra. $K_n$ be $O(n)$. I want to know any reference about the following statement.

For any irreducible admissible $(\mathfrak{g}_n\oplus \mathfrak{g}_m,K_n\times K_m)$ module $V$, there exist an irreducible admissible $(\mathfrak{g}_n, K_n)$ module $U$, and an irreducible admissible $(\mathfrak{g}_m, K_m)$ module $W$, such that $V$ is isomorphic to $U\otimes W$. Both $U$ and $W$ are uniquely determined by $V$ up to isomorphism.

Similar of this is true for representations of finite groups, smooth representations of p-adic groups. It seems to me that this is also true for real reductive groups, and I can't find it in any standard reference. I appreciate a lot if anyone would provide some related paper or book.

Is a space generated by a single vector of finite length in the p-adic supercuspidal GL(n) case?

2011-07-25T12:38:45Z

Given an irreducible supercuspidal representation $(\pi,V)$ of GL(n), embeds GL(n-1) into GL(n) on the left upper corner. Consider the restriction of $\pi$ to GL(n-1). I want to ask may I find some vector $v\in V$, such that the space generated by translations of $v$ under GL(n-1) is a representation of finite length of GL(n-1)?

Answer by unknown (google) for Does every normal subgroup appear as a kernel of an irreducible representation?

2011-04-12T13:23:26Z

Yes, consider the quotient by this normal subgroup, embeds it into symmetric group of $n$ letters with $n$ large enough, identify symmetric group with a subgroup of $GL(n)$, you get the representation.

Difference between action of group element and Lie algebra element in smooth representation

2011-03-29T14:47:20Z

Let $G$ be a real reductive Lie group, $P$ its parabolic subgroup with Levi decomposition $P=MN$, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Suppose given a smooth representation $(\pi,V)$ of $G$ with $V$ some Frechet space. We can form both $V/(\mathfrak{n}V)$ and $V/(N.V),$ where
$$ N.V=\operatorname{span}\{n.v-v|v\in V, n\in N\} $$

(here we take the closure of $\mathfrak{n}V$ and $N.V$ in the two quotients). Now question is there any difference between these two quotients? or equivalently, what's the relation between the closure of $\mathfrak{n}V$ and that of $N.V$? On one direction we have $\mathfrak{n}V$ is contained in $N.V$, so the question amounts to ask whether there is some $V$, so that the latter contains the former properly.

A related phenomenon is that, when considering the minimal parabolic subgroup $B=TU$, if $\chi$ is a generic character of $U$, $\eta$ the derivative of $\chi$, so $\eta$ is a nondegenerate complex linear form on the Lie algebra $\mathfrak{u}$ of $U$. There are two versions of Whittaker functional in literature, defined either in terms of pair $(U,\chi)$ or $\mathfrak{u},\eta$. And I'm wondering if they are equivalent.

Edit (Victor Protsak): definition of N.V has been made explicit.

reference containing the list of irreducible finite dimensional representation of real general linear group

2011-03-28T02:14:02Z

It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\mathfrak{g},K)$ module, and we do have a classification theorem in terms of highest weight, but it is not obvious to get the explicit constructions from this,especially in higher rank case.

So I'm wondering is there some good reference containing explicit constructions of all irreducible finite dimensional representations of $GL_n(\mathbb{R})$?

A naive question about composition factor of a representation

2011-03-19T06:45:00Z

Let $G$ be a Lie group, and $(\pi,V)$ is a continuous representation of $G$ which has finite composition series. A question I have which might be somehow naive is that: for any irreducible representation $(\sigma,W)$ of $G$, is it true that $(\sigma,W)$ occurs as one composition factor if and only if the set $Hom_G(V,W)$ is nonzero?

I have no idea how difficult or how easy this question might be, and any reference or answer is appreciated.

Edit: Thanks a lot for all of your answers, comments and examples. Now if $G$ is real reductive, $(\pi,V)$ is smooth admissible. Is there a way to determine all of the composition factors of $V$?

exactness of n homology functor

2011-03-01T08:51:58Z

Let $G$ be a real reductive group, $P=MN$ a parabolic subgroup with its Levi decomposition, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Now given a smooth representation $(\pi,V)$ of $G$, I wanna ask when the functor $H_0(\mathfrak{n},V)$ is exact? We may assume the representation $\pi$ has good properties, e.g., it is of moderate growth, $V$ is a nuclear Frechet space.

In general, the $\mathfrak{n}$ homology functor is only right exact on category of $\mathfrak{n}$ modules. So we may ask when this functor is exact if restricted to a subcategory.

Thanks.

Definition of L-function attached to automorphic representation

2010-03-28T04:47:44Z

Suppose $\pi$ is an irreducible automorphic representation of a reductive connected algebraic group $G$ over $\mathbb{A}_K$, here $K$ is a number field, $\mathbb{A}_K$ denotes its adeles. We have a restricted tensor product decomposition of $\pi=\otimes\pi_v$, where $\pi_v$ is an irreducible admissible representation for $G(K_v)$, and for all but finitely many $v$, $\pi_v$ is unramified.

We know how to define local L-factors at $v$ is $\pi_v$ is unramified, and we also know how to define local L-factors at archimedean places because of Langlands classification. So the question is how to define L-factors at ramified places?

As far as I know, at least for $GL_n$, we can define it as the gcd of some family of integrals via integral representation of L-function.

open orbits and invariant distributions

2010-11-26T00:35:20Z

Suppose $G$ is a p-adic algebraic group, $P=MN$ a parabolic subgroup of $G$ with its Levi decomposition, $\sigma$ be a irreducible representation of $M$, we use $I(\sigma)$ to denote the unique quotient of the normalized parabolic induction from $P$ to $G$. Now let $H$ be a closed subgroup of $G$.

The question I want to ask is that what's the relations between the existence of open-$H$ orbits in $P\G$ and $H$-invariant distributions on $I(\sigma)$, especially when we take $P$ to be the minimal parabolic subgroup?

Also I'm also wondering what's happening in the archimedean case.

Any answer or reference is welcome. Thanks.

reference help needed on a fact about poles of L-functions

2010-10-03T14:02:55Z

Suppose $\pi$ and $\rho$ are cuspidal automorphic representations on $GL(n)$ and $GL(m)$ respectively. Then the L-function $L(s,\pi \times \rho)$ has a pole iff and $m=n$ and $\pi$ is isomorphic to the contragradient of $\rho$ by some twist. Does anyone know some reference containing the proof of this fact?

I checked Rankin-Selberg convolution paper by Jacquet-P.S-Shalika. It mentioned this result and said the proof would appear somewhere.

Many thanks.

Questions on orbit properties of group action on varieties

2010-06-03T01:46:37Z

Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions concerning properties of orbits.

If there are only finitely many $G$ orbits, then in p-adic case, there exists open orbits. Now we may ask when there is exactly one open orbit, and what's the real case? What about infinitely many orbits?
what's the relation between the analytic topology and zariski topology on orbits? For example, if an orbit is closed in analytic topology, then it is also closed in zariski topology(over algebraic closure).
How to parametrize those orbits?
How to characterize open and closed orbits?

These obviously are difficult questions in general, and feel free to answer in any special case, like actions on flag varieties.

edit: As the above questions are too broad to answer. So let's try to focus on two special cases here most interested to me: $G$ a reductive group, $H$ a subgroup of $G$ which is the fixed points of some involution of $G$.

Case 1: $H\times H$ acts on $G$, one acts on left, the other on right.

Case 2: If $P$ is a parabolic subgroup, $H$ acts on $G/P$.

And we mainly concern question 1 and 4.

How to fit res map into a long exact sequence?

2010-05-25T03:27:52Z

Let G be a finite group, H a subgroup and V a G-module. Then the embedding H in G induces a restriction map on $H^{n}(G,A)$ to $H^{n}(H,A)$. My question is that is there any long exact sequence which contains this map? And generally how to compute $H^n(G,A)$ effectively when n is small, like 0,1.

what information of a representation was killed by Jacquet functor?

2010-04-27T01:00:31Z

Suppose $V_1$ and $V_2$ are two $(g,K)$ modules of some reductive group $G$ with maximal compact $K$. Let $P$ be the minimal parabolic of $G$, $U$ its unipotent part, and $u$ its Lie algebra. Suppose the quotients $V_1/uV_1$ and $V_2/uV_2$ are isomorphic as modules for the Levi component of $P$, then what else do we need to know to conclude that $V_1$ and $V_2$ are isomorphic as $G$ modules?

edit:Thanks for Kevin and Emerton's comments and sorry for the confusion about the base field. Here I'm assuming REAL reductive group.

reference for system of first order partial differential equations

2010-04-11T02:24:34Z

I want to know something from basic to recent results on solutions of first order partial differential equations (on manifold, or simply R^n), like existence, uniqueness and regularity of (weak) solutions (locally and globally), could any one recommend some references? I know basic stuff of analysis and differential equations. Thanks.

a question about irreducibility of representations and Kirillov conjecture

2010-03-30T02:38:33Z

Let $G=GL(\mathbb{R})$, $P$ be the subgroup of $G$ consisting of elements with the last row $(0,0,...,1)$. Then Kirillov conjecture states that for any irreducible unitary representation of $G$, its restriction to $P$ remains irreducible. This conjecture has been proved (not only over $\mathbb{R}$, but also over $\mathbb{C}$ and p-adic fields). Here I'm wondering if we consider irreducible smooth representations in Hilbert space(or Banach, Frechet space), does this conjecture remains true?

Another related question is generally, how to prove the irreducibility for a smooth representation besides the definition?

missing Lie algebra action on vectors in p-adic case?

2010-03-23T03:23:52Z

Suppose $(\pi, V)$ is a continuous representation of a real reductive group $G$, then the Lie algebra of $G$ act on the smooth vectors of $V$, and the $(g,K)$-modules are built from here.

It seems that we don't discuss the Lie algebra action on the representation space in p-adic group case. Is there any simple reason for that missing analog?

Comment by

2013-06-12T04:27:47Z

By the way, how to realize supercuspidal representations in $L^2$ space canonically? Would you please recommend some references?

Comment by

2013-03-27T04:17:38Z

@Qiaochu, I really mean real reductive Lie groups. Sorry for the ambiguity.

Comment by

2013-03-27T04:17:28Z

@Amritanshu,thanks for the link.

Comment by

2012-12-31T10:25:27Z

thanks a lot, this clarifies my question.

Comment by

2012-12-29T02:08:16Z

the precise question about applying thm 1.3.4 is that what are the weakly convergent eigenfunctions? The first n-terms in Taylor expansion?

Comment by

2012-12-28T05:26:37Z

I don't think it is a misprint, if you look at the bottom of page 31.

Comment by

2012-12-27T12:42:53Z

Thanks, and I understand your comment. But I'm still confused how to apply theorem 1.3.4 in Kubota's book?

Comment by

2012-10-28T00:14:20Z

thank you very much!

Comment by

2012-10-27T15:55:04Z

Thanks a lot for your explanation.Now if I'm interested in the cusp forms and Eisenstein series on full modular group, what are the better estimates for Hecke eigenforms? Any reference or answer is appreciated!

Comment by

2012-01-20T17:51:43Z

GH:Thanks a lot.

Comment by

2012-01-16T12:00:38Z

Alain: Of course we can always add zero seminorm. And the question I asked is that among all the seminors there, trivial or nontrivial, can we just pick up one particular one , and check the condition for it, then conclude that $\pi(f)v$ exists in $V$? Or we need to check the condition for all seminorms?

Comment by

2012-01-16T11:57:05Z

Alain: Thanks for your comment. It should be $\pi(f)v$ converges to a vector in V for any $v\in V$. Sorry for the confusion.

Comment by

2012-01-16T02:33:38Z

Thanks a lot. So it looks like the question for the particular case when $\pi$ is from a unitary representation I asked is fine.

Comment by

2012-01-16T00:57:44Z

Thanks. But it seems in that theorem it requires f to be compactly-supported...

Comment by

2012-01-15T17:43:36Z

Thanks for your comment. Here I'm more interested in if $\pi(f)$ converges to certain vector in $V$. Do we need the continuity prior to this question?