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Международная научная конференция "Гармонический анализ на однородных пространствах, представления групп Ли и квантование"
Международная научная конференция "Гармонический анализ на однородных пространствах, представления групп Ли и квантование" была проведена в Тамбове на базе Тамбовского государственного университета имени Г. Р. Державина с 25 по 29 апреля 2005 года. Она была организована кафедрой математического анализа университета. В работе конференции приняли участие 58 человек - как из-за рубежа (Франция, Голландия, США), так и различных городов России (Тамбов, Москва, С.-Петербург, Петрозаводск, Оренбург, Белгород, Ижевск, Сыктывкар, Волгоград). Заседания конференции посещали аспиранты и студенты университета.
Гармонический анализ на однородных пространствах - часть современной математики (функционального анализа), чрезвычайно актуальная и имеющая важные приложения в теоретической физике (в частности, квантовой механике). Он развивается многими математиками в России и за рубежом (Франция, США, Нидерланды, Германия, Япония, Дания, Швеция). Группа математиков Тамбовского университета занимает в этом ряду достойное место.
Цель конференции состояла в дальнейшем развитии этого перспективного и имеющего богатые приложения раздела математики, укреплении и расширении связей и сотрудничества между учеными России и других стран.
Тематика докладов, заслушанных на конференции, была достаточно обширной, она относилась к современным разделам математики и теоретической физики (программа Гельфанда-Гиндикина, квантования, канонические и граничные представления групп Ли, представления алгебраических структур, квантовые группы, вполне интегрируемые системы, формулы Планшереля и теоремы Пэли-Винера, спецфункции и др.).
Конференция была посвящена 70-летию двух замечательных российских математиков: Д.П.Желобенко (Российский университет Дружбы Народов, Москва) и В.М.Тихомирова (Московский государственный университет имени М.В.Ломоносова). Мы были рады видеть их в числе участников. Они оба выступили с докладами.
Конференция была поддержана Российским фондом фундаментальных исследований (грант № 05-01-10038г), Министерством образования и науки РФ (Ведомственная научная программа "Развитие научного потенциала высшей школы", проект № 380), Федеральным агентством по науке и инновациям и Тамбовским государственным университетом имени Г. Р. Державина.
Труды конференции печатаются в предыдущем и настоящем выпусках Вестника Тамбовского университета. Серия: Естественные и технические науки.
Сопредседатель Оргкомитета В.Ф. Молчанов
Conference “Harmonic analysis on homogeneous spaces, representations of Lie groups and quantization”
The conference “Harmonic analysis on homogeneous spaces, representations of Lie groups and quantization” was held in April 25 - 29, 2005, Tambov (Russia) at G.R. Derzhavin Tambov State University. It was organized by the chair of mathematical analysis of the university. Participants (58 people) came both from Russia (Tambov, Moscow, S.-Peterburg, Petrozavodsk, Orenburg, Belgorod, Izhevsk, Syktyvkar, Volgograd). Sessions of the conference were also attended by students and postgraduate students of the university.
Harmonic analysis on homogeneous spaces is a part of modern mathematics (of functional analysis), very actual and having important applications to theoretical physics (in particular, quantum mechanics). It is being developed by many researchers in Russian and abroad (France, USA, the Netherlands, Germany, Japan, Denmark, Sweden). A group of mathematicians from the Tambov University is known for its significant contribution in this research.
The work of the conference was aimed at enlarging future prospects of this part of mathematics with its most promising applications, strengthening and extending of connections between researchers from Russia and other countries.
The theme of the talks delivered at the conference were wide and various, they concerned modern parts of mathematics and theoretical physics (the Gelfand—Gindikin program, quantizations, canonical and boundary representations of Lie groups, representations of some algebraic structures, quantum groups, integrable systems, Plancherel formulas and Paley—Wiener type theorems, special functions etc.).
The conference was devoted to the 70-th anniversary of two remarkable Russian mathematicians: D.P.Zhelobenko (Russian University of People Friendship, Moscow) and V.M. Tikhomirov (M.V.Lomonosov Moscow State University). We were very glad to welcome them as the participants of the conference. They both gave talks.
The conference was supported by the Russian Foundation for Basic Research (grant № 05-01-0038g), the Ministry of Education and Science of Russian Federation (the Scientific Program “Development of scientific potential of higher school”, project № 380), the Federal Agency on Science and Innovations, and G.R.Derzhavin Tambov State University.
Proceedings of the conference are published in the preceding and this issues of the journal “Vestnik of Tambov University. Series: Natural and Technical Sciences”.
Co-chairmen of the Organized Committee
V.F.Molchanov
COMPLEX GENERALIZED GELFAND PAIRS
S. Aparicio and G. van Dijk University of Leiden, The Netherlands
Abstract
In this article we show that the pairs (SO(n, C), SO(n-1, C)) are generalized Gelfand pairs for ii ^ 2.
1 Introduction
Let G be a unimodular Lie group, H a closed unimodular subgroup and let X = G/H. The group G acts on the space of distributions on X, denoted by V'(X). A continuous unitary representation i of G on a Hilbert space H is said to be realizable on X if there exists a G-equivariant continuous linear injection j : % —> V(X). The pair (G,H) is called a generalized Gelfand pair if for all representations 7r with the above property, the commuting algebra of 7r(G) in the algebra End('H) of all continuous linear operators of 'H into itself, is abelian. This definition generalizes the classical notion of Gelfand pair, where H is assumed compact. A direct consequence of being a Gelfand pair is the multiplicity free decomposition of L2(X) into irreducible factors.
G. van Dijk, M. T. Kosters, W. A. Kosters and M. Poel have studied several real semisimple symmetric pairs of rank one in [5], [7], [8]. They have shown that all the non-Riemannian pairs are generalized Gelfand pairs, except the pairs (Spin(l,q + 1), Spin(l, q)) for q ^ 1, see [4]. G. van Dijk and E. P. H. Bosman also studied the p-adic analogues of some non-Riemannian pairs of rank one and they proved that they are generalized Gelfand pairs.
In this article we show that the pairs (SO(n, C), SO(n — 1 ,C)) are generalized Gelfand pairs for n ^ 2. This result is crucial for showing that every Hilbert subspace of the space of tempered distributions S'(Cn) invariant under the oscillator representation of SL(2,(7) x SO(n, C), decomposes multiplicity free, see [1].
We could also show that the pairs (SL(n, C), GL(n — 1, C)) and (Sp(n, C), Sp(n — 1, C) x Sp(l,C)) are generalized Gelfand pairs for n ^ 3 applying a similar method as for the real case in [7] and [8]. The difference is that we had to introduce two differential operators instead of only one. We do not include the proof in this article.
2 Definition of Generalized Gelfand Pairs
We shall give a brief summary of the theory of invariant Hilbert subspaces and generalized Gelfand pairs, for more details see [5]. Let G be a Lie group and H a closed subgroup of G. We shall assume both G and H to be unimodular. Denote by T>(G), V(G/H) the
space of C°°-functions with compact support on G and G/H respectively, endowed with the usual topology. Let V'(G), V'(G/H) be the topological anti-dual of V(G) and V(G/H) respectively, provided with the strong topology.
A continuous unitary representation n of G on a Hilbert space % is said to be realizable on G/H is there is a continuous linear injection j : H —> V'(G/H) such that
jir{g) = Lgj
for all g G G (Lg denotes left translation by g). The space j(H) is called an invariant Hilbert subspace of V'(G/H). We shall take all scalar products anti-linear in the first and linear in the second factor.
Definition 1 The pair (G, H) is called a generalized Gelfand pair if for each continuous unitary representation ir on a Hilbert space 'H, which can be realized on G/H, the commutant of n(G) in the algebra End(H) of all continuous linear operators ofT-L into itself, is abelian.
For equivalent definitions we refer to [3] and [10]. A large class of examples is given by the Riemannian semisimple symmetric pairs and by the nilpotent symmetric pairs [3], [2].
A usefull criterion for determining generalized Gelfand pairs was given by Thomas ([10], Theorem E). We shall apply it throughout this paper. Its proof is easy and straightforward (I.e.).
Denote by V'(G,H) the space of right if-invariant distributions on G provided with the relative topology of V'{G). It is well-known that V(G,H) can be identified with V'(G/H).
Criterion 2.1 Let J : T>'(G,H) —> V'(G,H) be an anti-automorphism. If J% = H (i.e. (J\H) anti-unitary) for all G-invariant or minimal G-invariant Hilbert subspaces ofT>'(G, H), then (G, H) is a generalized Gelfand pair.
We shall apply it in the following form.
Criterion 2.2 Let r be an involutive automorphism of G which leaves H stable. Define JT = TT for all T G U'(G,H). If JT = T for all bi-H-invariant positive-definite (or extremal positive-definite) distributions on G, then (G,H) is a generalized Gelfand pair.
Remark 1 TT is defined by < TT, f >=< T, fT > (/ e T>(G)) and fT(g) = f{r(g)) (g G G). T is defined by < f, / >=< T, / > (/ e V{G)).
An important consequence of being a generalized Gelfand pair is the multiplicity-free désintégration of the left regular representation of G on L2(G/H). So one could, more or less without ambiguity, call this the Plancherel formula for G/H. If a fixed parametrization is used for the set of irreducible unitary representations realized on G/H, there is no ambiguity at all.
Let Z denote the algebra of all analytic differential operators on G which commute with left and right translations by elements of G. Any bi-H-invariant common eigendistribution of all elements of Z is called a spherical distribution. It is a well-known consequence of Schur’s Lemma that any bi-H-invariant extremal positive-definite distribution on G is spherical. Spherical distributions play an important role in the harmonic analysis on G/H.
3 Morse’s Lemma
Let X be a complex analytic manifold of dimension n (n 6 N), and / : X —> C an analytic function on X. The tangent space of X at a point a;0 will be denoted by TXxo. A point x° G X is called a critical point of / is the induced map /* : TXxo —> TCf(xo) is zero. If we choose a local coordinate system (x\,... ,xn) in a neighborhood U of x° this means that
A critical point x° is called non-degenerate if and only if the matrix
d2f
dxidxj
(*°)
is non-singular.
For a critical point x of / let the Hessian Hxf of / at x be the quadratic form on the tangent space TxX which is defined by
Hxf eCn’
\i=1 / J
in local coordinates (cci,..., xn) at x.
Theorem 3.1 (Morse’s lemma) Let f : X —i C be an analytic function from a complex manifold X into C and let x° be a non-degenerate critical point of f. There are local coordinates (xy,..., xn) at a;0 with (0,...,0) corresponding to x° such that f can be written
f(x i,... ,xn) = f(x°) + x\ + ---+x\.
The proof of the theorem is similar to the proof of Morse’s lemma in [6] p. 146.
4 The pairs (SO(n, C), SO(n - 1, C))
Assume n ^ 3.
Let G = S0(n,C), H = SO(n — 1 ,C)). The space X = G/H can clearly be identified
with the set of all points x = (x\,, xn) in Cn satisfying x\ H-----------+ x\ = 1.
We consider the following function Q on the space X which parametrizes the if-orbits on
X:
Q(x) = x i.
Q is an ff-invariant complex analytic function on X with Q(x°) = 1.
Define X(z) — {x E X\ Q{x) = z} for z € C. Now the if-orbit structure on X is as follows:
Lemma 4.1 a) Let z E C, z ^ 1,-1. Then X(z) is a H-orbit.
b) X(l) consits of two H-orbits: {a:0} and Ti = X(l) \ {a;0}.
c) X(—1) consits of two H-orbits: {—a;0} and T_i = X(—1) \ {—a;0}.
In order to treat the sets X(l) and X(—1) separately, we choose open H-invariant sets X~\ and Xi such that X(-l) C X_i, X(l) <jL X_i, X(l) C Xu X(-l) £ Xx and X^UXi = X. These sets clearly exist.
The critical points of Q are x° and —a;0. Both critical points are non-degenerate.
We examine Q in the neighborhood of a critical point. Firstly, near a:0 there exists a coordinate system {u>i,..., wn^ 1} such that
This is due to Morse’s lemma.
From the properties of Q, we deduce applying [9] the existence of a linear map M, which assigns to every f G V(X) a function Mf on C such that
for all F G V(C). Here dx is an invariant measure on X, dz = dxdy {z = x + iy). Mf(z) gives the mean of / over the set X(z). Let Ti = M(V(X)) and Hi = M(V(Xi)) (i = —1,1). Using the nature of the critical points of Q and the results of [9], §6 we get:
Q(wi,wn-i) ~l + w\-\------+ wl_x,
x° corresponding to (0,..., 0). The Hessian HxoQ at a:0 is given by
HxoQ(wu ... ,wn-i) = -wj-----------wl-v
Secondly near —a:0 there exists a coordinate system ... ,wn-i} such that
Q(wi,.. .,wn-1) = -1 + w\ H--------+ Wn-n
—a:0 corresponding to (0,..., 0). The Hessian H_xoQ at — x° is given by
U = {(р + г)офо + 77iV>i| <р,Фо,Ф\ e T>{C)} 'H-i = {<po + тФо\ т,Фо e
= (<^i +гцфг\ (риФх e V(Q(Xi)},
where
( \z + l\n 2 if n is even
I \z + l|n-2Log|z + 1| if n is odd
if n is even
and
if n is even Log|z — 1| if n is odd
If we topologize H, Ti-i and Hi as in [9] we have for * = —1,1:
a) M : V{Xi) —> Hi is continuous.
b) The image of the transpose map M1 : 7-L[ —y V{Xi) is the space of H-invariant distributions on Xi. M’ is injective on because M is surjective.
Similar properties hold for M :V(X) —> H.
So, given an //-invariant distribution on X, there exists an element S G %' such that
<T,<p>=< S,Mip> (4.1)
for all ip G V{X) . Fix Haar measures dg on G and dh on H in such a way that dg = dxdh, symbolically. For / G V{G) put
fHx) = [ f(9h)dh (x = gH).
Jh
Given a bi-//-invariant distribution To on G, there is an unique //-invariant distribution T on X satisfying < To, / >=< T,p > (/ G V(G)). This is a well-known fact.
We are now prepared to prove that (G,H) is a generalized Gelfand pair. We apply Criterion 2.2 with JT = T (T 6 V(G,H)). We have to show that T = T for all bi-H-invariant positive-definite distributions T on G. Since T = T for such T, we shall show the following: for any bi-iT-invariant distribution T on G one has T = T. Here < T, / >=< T,f >, f(g) = /(<7_1) (g G G, f £ V(G)). In view of the relation between bi-fi-invariant distributions on G and //-invariant distributions on X, and because of (4.1), this amounts to the relation
= M(/#)
for all / G T>{G). For all F G T>(C) one has
f F(z)M[(f)*}{z)dz = j G
Since Q(g) = Q(g :) (g G G) we get the result.
So we have shown:
Theorem 4.1 The pairs (SO(n, C), SO(n — 1, C)) are generalized Gelfand pairs for n ^ 3.
The case n = 2 is easily seen to provide a generalized Gelfand pair too, since SO(2, C) is an abelian group.
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