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Home , Algebraic group, Arithmetic group, Class of groups, Discrete group, Gopal Prasad, Simple Lie group

Lie groups and algebraic groups

M S RAGHUNATHAN and T N VENKATARAMANA∗ School of , Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India. ∗e-mail: [email protected]

We give an exposition of certain topics in Lie groups and algebraic groups. This is not a complete survey and is meant to highlight certain aspects with which the authors are familiar, and to which there has been a significant contribution from working in India.

1. Introduction linear classical groups GLn(R), GLn(C), SOn(R), SOn(C), Spn(R)andSpn(C). Let us call a con- One of the fundamental branches of mathematics nected Lie simple if G has no connected is the study of symmetries of any mathemati- normal closed other than G and the cal object (like a geometric object, or a differ- identity. It is a remarkable fact that simple ential/polynomial equation, etc.). The symmetries Lie groups can be completely classified; they form a group, and group theoretical properties of are the special linear groups, orthogonal groups this group give us information on other mathemati- and symplectic groups. Apart from these, the cal properties of the object under consideration. list is a finite one (the so-called exceptional For example, the solvability of the ‘Galois group’ groups). This is the Cartan–Killing classification, of a polynomial equation is equivalent to the solva- whichnowadays,isdescribedintermsofthe bility of the equation by radicals. A similar state- ‘Dynkin diagrams’. ment can be made for solvability of differential A related of groups are simple algebraic equations. groups. They can be defined over an arbitrary field. The most ubiquitous is that of Lie For a discussion on classification of these groups, groups. These are manifolds (geometrical objects, see section 5. functions on which can be differentiated) such that the group operations are infinitely differentiable. Lie groups occur naturally in geometry, analysis, 3. algebra and physics. Prototypical examples are the groups GLn(R)(GLn(C)) of invertible n×n matri- 3.1 Finite dimensional representations ces with real (complex) entries. By its very nature, a has analytic, geo- Lie groups are essentially (up to coverings) closed metric and algebraic properties, each of which is subgroups of GLn(R). One of the fundamental a highly developed field of study. These fields are problems in the theory of Lie groups was to find all interrelated and each of these fields contributes to possible ways in which a Lie group can be (embed- the other. ded as) a of GLn(R)(orGLn(C)). This is a problem which is completely solved for connected 2. Examples and classification Lie groups. The foregoing problem is equivalent to finding We first give some examples of Lie groups. all possible group homomorphisms G → GLn(C)– The most frequently occurring ones are the with varying n – which are infinitely differentiable

Keywords. Lie groups; representations; algebraic groups; arithmetic groups, rigidity.

627 628 M S RAGHUNATHAN AND T N VENKATARAMANA

(smooth). Such homomorphisms are called (finite representations. The explicit realisation is due to dimensional) ‘representations’ of G.Inother Borel and Weil. words, elements of G may be viewed as invertible An important problem in physics is the decom- linear transformations of the vector space Cn.By position of a representation ρ of a compact Lie an abuse of notation, the vector space Cn together group restricted to a compact subgroup H.In with the G-action, is also called a representation particular, the decomposition of a tensor product of G. is especially important; the PRV conjecture (see We will describe – briefly – the solution of this [12]) describes exactly what happens in this case problem, for compact connected Lie groups. A com- for certain important constituents of the tensor pact Lie group G may be shown (with some work, product; the conjecture was proved by Shrawan using the theory of compact operators) to be none Kumar [25]. In a different direction, a unique fac- other than a closed subgroup of the torisation theorem for tensor products was proved t U(n)ofn × n-matrices g ∈ GLn(C)with gg =1, by C S Rajan [22]. for some n.IfG is connected, it is possible to replace G by a conjugate in U(n) such that the 3.2 Infinite dimensional representations intersection with G of the group of diagonals in U(n) is a maximal connected abelian subgroup T We will now consider unitary representations of G of G (it is called a ‘maximal torus’ in G). All maxi- i.e. infinite dimensional Hilbert spaces H on which mal tori in G are conjugate (this purely algebraic non-compact Lie groups G act preserving the inner statement was first proved by using the Lefschetz product. A special class of Lie groups are those fixed point theorem, which is a fundamental whose is a direct sum of irreducible result in algebraic ; this is an instance (finite dimensional) representations of G each of of geometric properties of G yielding algebraic which has strictly greater than one. Such information). groups are called semi-simple Lie groups. It is no Direct sums and tensor products of representa- longer true that a representation is a direct ‘sum’ tions may be defined in an obvious way. A repre- of irreducibles. The direct sum has to be replaced sentation is said to be irreducible, if it cannot be by a direct integral. written as a direct sum of two representations. If A basic example of a representation is the space 2 G is compact, every representation can be written L (G) of the space of square integrable functions as a direct sum of irreducible ones. Therefore, the on G (square integrable with respect to the Haar problem of describing representations reduces to measure). Even in this case, a description of the that of irreducible representations. It can be shown direct integral decomposition is a formidably diffi- that if ρ : G → GLn(C) is irreducible, the trace cult problem. It was however, completely solved by function g → trace(ρ(g)) on G completely deter- the great Indian Harish-Chandra in mines the representation. aseriesofpapers. To continue, one may show that the quotient The description of representations which occur, manifold G/T is a compact complex manifold such together with the number of times they occur (the that the group G (which acts on G/T by trans- multiplicity, or the Plancherel measure in the case lations), acts by holomorphic automorphisms. If χ of direct integrals) in the direct integral decomposi- 2 is a character of T (a continuous homomorphism tion of L (G) is called the Plancherel theorem. The of T into U(1) = the ), then one may classification of these representations contributing form a ‘line bundle’ Lχ on G/T which can be to the Plancherel theorem is in terms of fundamen- shown to be a holomorphic line bundle. The space tal building blocks, which are called ‘discrete series 0 H (G/T, Lχ) of holomorphic sections can be shown representations’. Harish-Chandra proved that a to be finite dimensional, on which G acts by left semi- G has discrete series if translations. One has the fundamental theorem of and only if G has a connected maximal abelian , and Andre Weil: subgroup T which is compact (a compact Car- tan subgroup T ). Harish-Chandra gave a formula Theorem 1. The representation ρ = ρχ = (the Plancherel measure) for the representations, 0 2 H (G/T, Lχ) is irreducible. Moreover, every irre- together with multiplicities, which occur in L (G). ducible representation of G may be realised in this The Plancherel measure for G/K (K amaxi- way, for some character χ of T . The trace function mal compact subgroup of G) was computed for trace(ρ(g)) can be completely described in terms many classical groups by T S Bhanu-Murthy [4]. of χ. The complete classification of the representations occurring in L2(G) was achieved by Knapp and Historically, Hermann Weyl first described the Zuckerman [5]. irreducible representations in terms of their Somewhat similar to the case of compact Lie traces, without an explicit construction of the groups, Harish Chandra gave a formula – without LIE GROUPS AND ALGEBRAIC GROUPS 629 explicitly constructing the discrete series rep- of projective space which preserves its ‘geometry’ resentations – for the ‘trace distribution’ of (i.e. incidence relations such as subspaces of a given discrete series. The problem of explicitly con- vector space) is actually a linear isomorphism. The structing the discrete series was solved by various proof is a repeated use of the high transitivity people. Many Indian mathematicians contributed of the action by the group of linear isomorphisms to this development: an algebraic construction was on the ‘geometry’ of the projective space. Similarly given by K R Parthasarathy, Ranga Rao and theorems of affine geometry translate into pro- V S Varadarajan. A description, closer in spirit to perties of the affine transformation group. Similar the Borel–Weil (–Bott) theorem, was conjectured results hold for other geometries. by Langlands, and proved by M S Narasimhan– Felix Klein had the idea that extension of these Okamoto (in the case when the results to other geometries can be deduced from of the group G is a Hermitian symmetric domain) group theoretical properties of certain transfor- and by R Parthasarathy (for G arbitrary but the mation groups associated to these geometries. He weight of the discrete series is sufficiently regu- sketched a programme (the Erlangen programme) lar). A complete proof of Langlands’ conjecture wherein he conjectured that all of geometry may was given by Wilfried Schmidt. be ‘reduced’ to . The programme was The quotient G/T has many different complex partly successful. structures which are G-invariant. Fix one. Fix a character χ of T . One obtains a holomorphic 4.2 Tits buildings line bundle Lχ over G/T analogous to the com- pact situation. Consider the so-called ‘Dolbeault completely turned this situation complex’ of square summable differential forms around and proved a beautiful generalisation of with coefficients in Lχ on the complex mani- the fundamental theorem of . fold G/T ,andforp ≥ 0, let ρ = ρχ,p be the He associated a certain geometry to groups G p-th cohomology group of this complex, consi- over fields k, and showed that for groups of k- dered as a G-module. Then Langlands’ conjecture rank ≥2, maps between these ‘Tits buildings’ arose (Schmidt’s theorem) is the following (compare with from group homomorphisms; so theorems for these theorem 1). groups (such as finding generators and relations) were deduced from geometric properties of the Theorem 2. For exactly one value of p,the buildings. Here, the k-rank of an representation ρ is an irreducible discrete series G over a field k is the dimension of a maximal k- representation. Moreover, every discrete series re- subgroup of G,whichisk-isomorphic to a product presentation arises this way. of copies of GL1. This important group-theoretic notion occurs repeatedly in the sequel. 3.3 Unitarizability Generalisations of symmetric spaces were considered by Bruhat and Tits, and they con- The problem of describing all unitary irreducible structed p-adic symmetric spaces – the ‘Bruhat– representations of a semi-simple Lie group G is, as Tits buildings’. The Bruhat–Tits buildings have yet, not completely solved. For all classical groups, had applications in many fields: in the theory it is known by recent work of Vogan, Barbasch of automorphic forms, where local zeta func- and others. The difficulty is in showing that cer- tions of Shimura varieties are related to proper- tain irreducible representations of the Lie algebra ties of an associated Bruhat–Tits buildings, for have an (G-) invariant positive definite Hermitian cohomology computations in the series of papers form. This is the question of ‘unitarizability’ of by Raghunathan and G Prasad, in questions representations of the Lie algebra and there has on classification of bundles by Raghunathan and been a huge collaborative work recently on obtain- Ramanathan, and in the work of Garland and ing effective algorithms for deciding when the Lie Raghunathan on loop spaces. algebra representations do have such a Hermitian form. R Parthasarathy and Siddharth Sahi have contributed significantly to this problem. 5. Discrete subgroups of Lie groups

5.1 Rigidity 4. Geometry and groups We begin with some examples. A compact 4.1 The Erlangen programme of Klein Riemann surface of genus g ≥ 2maybeviewed as a discrete subgroup Γ of the Lie group SL2(R) Let us recall the fundamental theorem of projective of 2 × 2-real matrices with one. The geometry, which says that any set theoretic map discrete subgroup will further have the property 630 M S RAGHUNATHAN AND T N VENKATARAMANA that the quotient Γ\G is compact. Moreover, the general concept (as it will be needed later). Let k discrete subgroup is determined only up to conju- be a field and K an algebraic closure of k.LetG gacy. The discrete subgroup is simply the funda- be an algebraic subgroup of GLn(K) defined over mental group of the Riemann surface, which by the k (or simply a k-algebraic subgroup). This means uniformisation theorem, may be thought of as the that G is a subgroup of GLn(K), which is also the deck transformation group of the universal cover- set of zeroes of a collection of polynomials in the ing space (the upper half plane) of the Riemann entries of elements of GLn(K)andtheinverseof M surface. It can be shown that the space g of the determinant function on GLn(K), all of which Riemann surfaces of genus g, is a complex alge- have coefficients in k.Thek-points G(k)ofG is − braic space of dimension 3g 3 i.e. of real dimen- the intersection of G with GLn(k). sion 6g − 6. However, all these surfaces have the Suppose now that k is a global field (number same topology; hence the space of (non-conjugate) fields are global fields) and S is a (finite) set of homomorphisms of Γ into SL2(R) (which are injec- valuations of k containing all the Archimedean tive with discrete ) is a ‘continuous space’ of valuations of k. For a non-Archimedean valuation − dimension 6g 6. In particular, Γ may be continu- v of k,wedenotebykv the completion of k at v, R ously deformed in SL2( ) with the deformation and by Ov the ring of in kv.AlsoforS − space being (6g 6)-dimensional. as above, OS is the ring of S-integers in K: OS is R If SL2( ) is replaced by other (semi-) simple the set of all x in k which are in Ov for all non- R ≥ groups (e.g. SLn( )withn 3), then the space Archimedean v outside S.WedenotebyG(OS )the of deformations of a cocompact Γ subgroup of all elements x in G(k) such that x and may similarly be considered. However, repeated x−1 are both in GLn(OS ). A subgroup Γ of G(k) attempts of constructing such deformations failed, is defined to be an S-arithmetic subgroup of G if and the only known examples arose from arith- Γ ∩ G(OS ) has finite index in Γ as well as in G(OS ). metic. It was then conjectured by A Selberg that When S is precisely the set of Archimedean places, there were no continuous deformations of these an S-arithmetic subgroup will be called an arith- ‘higher rank’ discrete groups. This was proved by metic group. Selberg in many special cases and by Andre Weil For each v in S, the group G(kv) has a natural in general using methods of Calabi–Vesentini. This locally compact topology deduced from that on kv. lack of deformation may be viewed as saying that We denote by GS the direct product of the G(kv )as Γ is locally rigid: all nearby deformations are con- v runs through S. One then has a natural diagonal jugate to each other. The Weil rigidity theorem inclusion of G(k)inGS, and hence also of any S- was extended to non-cocompact discrete groups (of arithmetic subgroup Γ ⊂ G(k). With this notation rank one) by Garland and Raghunathan. we have the following fundamental result due to George Mostow discovered a much stronger Borel and Harish-Chandra. property for these Γ; he termed this ‘strong rigi- dity’: it says that if two cocompact discrete groups Theorem 3. Suppose that G is semisimple (this in two different simple Lie groups G and G are means that the Lie algebra of G is semisimple). isomorphic as abstract groups, then any abstract Then Γ has finite covolume in GS. isomorphism arises from an isomorphim of the ambient Lie groups G and G,providedG has When S is precisely the set of Archimedean dimension >3. This strong rigidity was extended valuations, GS is a real Lie group (kv is the field of to rank one non-cocompact discrete groups by real or complex numbers for all V in S), so that we [15]. The proofs of these results obtain in this fashion discrete subgroups of semi- started with the observation that a map of lat- simple Lie groups. tices gave rise to an equivariant quasi-isometry For certain choices of G and k, it turns out that between the symmetric spaces, which gave a map the projection of Γ on one factor G(kv )ofGS between the boundaries of these symmetric spaces. is a discrete cocompact subgroup of that factor. The boundary turns out to be the Tits building Borel used this to show that one can construct and therefore the theorem of Tits on maps between cocompact discrete subgroups in any semisimple buildings being induced from homomorphisms of real Lie group. the groups yields the strong rigidity theorem. One says that a discrete subgroup Γ of a semi- It had been noted (by Selberg, for instance) that simple Lie group H is arithmetically defined if the  the only general method of constructing cocom- following holds: let H be the quotient of H by its pact (more generally, finite covolume) discrete sub- centre. Then there is a number field k, a semisimple groups in higher rank groups was ‘by arithmetic’. algebraic group G over k and an arithmetic sub- In to explain that, we need to define the group Γ of G, such that H is isomorphic (as a real notion of an ‘arithmetic subgroup’ of an algebraic Lie group) with a factor of the product of G(kv ), group. We will in fact define a somewhat more where v runs through all Archimedean places of k; LIE GROUPS AND ALGEBRAIC GROUPS 631 further, the image in H of the discrete subgroup This vanishing result was greatly generalised ΓofH, and the projection of the by Matsushima for the trivial representation and  ⊂   Γ v G(kv)onH , are commensurable in H , Raghunathan in general: the result states that if i.e. their intersection has finite index in both. This Γ is a in a simple Lie group G other than seems a rather technical definition, but it is neces- SO(n, 1) or SU(n, 1), and ρ is any finite dimen- sary when working in the general context in which sional representation of G,then the results naturally find their place. Selberg and Pjatetskii–Shapiro conjectured that H1(Γ,ρ)=0. all discrete subgroups with finite covolume in a semi-simple Lie group are arithmetically defined Even in the case of SO(n, 1) and SU(n, 1) the (in the technical sense explained abvove). It was vanishing theorem holds, except for a special class proved for a fairly substantial class of discrete of ρ. groups (to be specific, for the class of dis- There is a general formula, called the Mat- crete groups Γ in groups with Q-rank ≥ 2) by sushima formula, which relates cohomology of M S Raghunathan, and proved by G A Margulis in cocompact discrete groups to the occurrence in full generality. L2(Γ\G) of certain ‘cohomological representations’ Margulis discovered an extremely strong of G. The construction and classification of these generalisation of rigidity (which was termed representations was almost completely solved by ‘’ by Mostow). It says that any linear Parthasarathy and Kumaresan. A much more gene- representation (over any locally compact field) of ral result was proved by Vogan and Zuckerman. the discrete group Γ extends essentially to the When the arithmetic discrete group is no longer whole Lie group G,providedG has R-rank at compact, the analogue of the Matsushima formula leasttwo–theR-rank of G is characterized as the becomes much more difficult to establish. Even to dimension of a maximal connected abelian sub- prove vanishing of cohomology, one must relate the group A of G, such that no non-trivial element of cohomology to suitable square summable forms. A is a element of G (unipotent elements This was accomplished in [19]. A recent theorem are those which act on the Lie algebra with all of Jens Franke shows that cohomology is repre- eigenvalues equal to 1). sented by automorphic forms. A description of This phenomenon of super rigidity was known – the ‘weighted L2-cohomology’ of a non-cocompact at least for arithmetic Γ obtained from Q-groups arithmetic group was achieved by Arvind Nair [10]. with Q-rank ≥2 – using the property (see section 5.4 below), by the work of 5.3 Normal subgroups Bass–Milnor–Serre and M S Raghunathan: in this case, the authors exploited the existence of unipo- Bass, Milnor and Serre discovered that if n ≥ 3, tent elements in Γ. then any infinite of SLn(Z)was Margulis showed, however, that super rigidity of finite index. This theorem was extended by holds for any higher rank lattice Γ; the proof Raghunathan to any non-cocompact arithmetic used ideas from diverse fields and a crucial com- irreducible subgroup of a semi- of ponent was the use of and measure R-rank greater than one. Using ergodic theoretic theory in establishing essentially algebraic proper- techniques, Margulis proved that if Γ is an irre- ties of lattices. Remarkably, he showed that the ducible lattice Γ in any higher rank semi-simple Lie argument can be turned around: that superrigidity group G, then all infinite normal subgroups are of implies the Selberg–Pjatetskii Shapiro conjecture finite index (extending the result of Raghunathan on arithmeticity. to the cocompact case). In the course of proving These results were extended to the positive this, Margulis proved that any Γ-equivariant mea- characteristic case by T N Venkataramana [27]. surable quotient of the variety G/P (P a mini- mal real parabolic subgroup) is actually G/Q with 5.2 Cohomology Q ⊃ P a parabolic subgroup. Dani proved a con- tinuous version of this result, answering a ques- The theorem of Weil referred to above may be tion of Margulis. Thus, classifying infinite normal stated differently. Denote by Hi(Γ,ρ)thei-th coho- subgroups is equivalent to classifying normal sub- mology group of the discrete group Γ with coeffi- groups of finite index in ‘higher rank’ arithmetic cients in a finite dimensional representation ρ.The groups. local rigidity theorem of Weil follows from the vanishing result (for the ‘’ 5.4 The congruence subgroup problem Ad): We now turn to classification of finite quotients H1(Γ, Ad) = 0. of arithmetic groups. Let us start with an 632 M S RAGHUNATHAN AND T N VENKATARAMANA example: the group Γ = SL2(Z)of2× 2-matrices The same assertion holds also for a number with coefficients and determinant one. This of anistropic G as well. For spin groups, this is has many subgroups of finite index; one way of due to Kneser. His techniques were generalised by obtaining finite quotients of SL2(Z)istocon- Rapincuk and Tomanov (independently) to cover sider an integer k ≥ 0 and the attendant ring groups of type Dn and some others. homomorphism Z → Z/kZ; this induces a homo- The centrality of C(S, G)isthemainthrust morphism of groups SL2(Z) → SL2(Z/kZ). The of [20] and [21]. The structure of C(S, G)is image is surjective and is a finite quotient. The then a consequence of some cohomology compu- consists of matrices, which are congruent tations due to Prasad and Raghunathan [16,17]. to the identity matrix modulo k, and is a sub- These last results are generalisations of theorems group Γ(k) of finite index. This is called a prin- about central extensions of p-adic split and quasi- cipal congruence subgroup of level k. A subgroup split groups due respectively to C C Moore and of SL2(Z) which contains a principal congruence V Deodhar. subgroup Γ(k) is called a congruence subgroup of B Sury solved the congruence subgroup problem SL2(Z). for a large class of S-arithmetic groups, where S is The question is whether every finite index sub- the complement of a finite set; this generalised a group of SL2(Z) is a congruence group; this turns theorem of Margulis. out to be false for SL2(Z) but holds true for groups like SL3(Z). One can ask a more general ques- tion. Let k be a global field and S a finite set 6. Algebraic groups, groups over of places including all Archimedean places. Let other fields OS be the ring of S-integers in k.LetG be a linear algebraic subgroup of GLn defined over k. 6.1 Classification The group G(k)ofk-points of G can be made into a by declaring the set of Most Lie groups are the groups of R-points of R- all S-arithmetic subgroups (e.g., section 5.1) as algebraic groups (e.g., section 5.1); some examples C C C a fundamental system of neighbourhoods of the are SLn( ), SOn( ), Spn( ). A classification of R identity. The completion of G(k)(withrespect k-algebraic groups is possible over fields k like , C to the left uniform structure) is a locally com- and local fields. Over number fields, the classifi- cation depends on the ‘Hasse principle’ which says pact group, which we denote by Ga in the sequel. G(k) can also be topologised by declaring the set essentially that if two groups are isomorphic over of all S-congruence groups to be a fundamental all completions, then they are isomorphic. The clas- system of neighbourhoods of the identity: an S- sification over other fields is not completely known. arithmetic subgroup is an S-congruence subgroup Over certain special fields, there is a conjecture of if it contains the kernel of the natural homo- Serre which describes the classification in terms of morphism of G(O )inGL (O /I), where I is vanishing of . Serre’s conjecture S n S was proved for all classical groups by R Parimala anon-zeroidealinOS. The corresponding com- pletion of G(k) is denoted G . One then has and Fluckiger. c A related question is the classification of prin- a natural map of Ga in Gc whose kernel we denote by C(S, G). Then C(S, G) measures the fail- cipal G-bundles and was settled by Raghunathan, ure of S-arithmetic subgroups to be S-congruence Ramanathan and in many special cases, by subgroups. Parimala. Much is known about C(S, G)forG isotropic Recently Suresh has proved the related conjec- over k. The following result is due to Raghunathan ture that all quadratic forms over certain fields [20,21], and subsumes earlier results due to represent zero. Bass–Serre–Lazard, Mennicke, Bass–Milnor–Serre, 6.2 Kac–Moody groups and algebras Matsumoto, Vasserstein and Bak–Rehman; [20] deals with the case of groups of k-rank 1, and makes We have already mentioned work on loop groups. use of earlier results due to Serre on SL2. These loop groups are closely connected to groups over rings of the form C[t, t−1] (the Laurent series Theorem 4. Assume that G is simply connec- ring in one variable. These are called Kac–Moody ted, isotropic over k and that the sum of the kv- Lie groups (with associated Lie algebras); their ranks of G as v varies over S is at least 2 (where kv representation theory has remarkable similarities is the completion of k at v). Then the group Ga is a with the finite dimensional case, and yet there are central extension of Gc,andC(S, G) is isomorphic subtle differences. Much pioneering work was done to a subgroup of the group of roots of unity in k (in by V Chari and A Pressley, and S E Rao has con- particular, it is a finite ). tinued work on these and related topics. LIE GROUPS AND ALGEBRAIC GROUPS 633

7. Connections with ergodic theory groups, (I) and (II); Ann. Math 116(2,3) 389–455, 457–501. [6] Kumaresan S 1980 Canonical K-types; Invent. Math. One of the most active areas of research is the inter- 59(1) face between Lie groups and ergodic theory. The 1–11. [7] Lang S 1975 SL(2, R), Addison Wesley Publishing Co., fundamental theorem which relates these two fields Reading, Mass.-London-Amsterdam. is the Howe–Moore ergodicity theorem. [8] Margulis G A 1984 Arithmeticity of lattices real rank greater than one; Invent. Math. 76(1) 93–120. Theorem 5. If ρ : G → U(H) is a representa- [9] Margulis G A 1989 Discrete subgroups and ergodic the- tion of a non-compact simple Lie Group G on a ory, , trace formulas and discrete groups Hilbert space H,andv ∈ H is a non-zero vector (Oslo), 377–398 (Boston: Academic Press). [10] Nair A 1999 Weighted cohomology of arithmetic whichisfixedunderanon-compactsubgroupU of 150(2) G, then all of G fixes the vector v. groups; Ann. Math. 1–31. [11] Okamoto and Narasimhan M S 1970 An analogue of the Borel–Weil–Bott theorem for Hermitian symmetric An immediate consequence is that if H = (2) 2 pairs of noncompact type; Ann. Math 485–511. L (Γ\G), then every non-compact closed subgroup [12] Parthasarathy K R, Ranga Rao and Varadarajan V S U of G acts ergodically on Γ\G. This theorem is 1967 Representations of complex semi-simple Lie already used crucially in the arithmeticity theorem groups and Lie algebras; Ann. Math. 85(2) 383–429. of Margulis alluded to above. In particular, a [13] Parthasarathy R 1972 Dirac operators and the discrete 96(2) unipotent subgroup U acting on Γ\G acts ergodi- series; Ann. Math 1–30. [14] Parthasarathy R 1978 A generalisation of the Enright– cally. As a consequence, almost all orbits of U are Varadarajan modules; Compositio. Math. 36 53–73. dense. [15] Gopal Prasad 1973 Strong rigidity for Q-rank one lat- A major driving force in the area was the con- tices; Invent. Math. 21 255–286. jecture due to Dani and Raghunathan, that orbit [16] Gopal Prasad and Raghunathan M S 1984 Topologi- closures of connected unipotent groups acting on cal central extensions of semisimple groups over local \ fields, I & II; Ann. Math. (2) 119(1) 143–201; ibid,(2) Γ G can be completely described (the last sen- 119(2) tence of the previous paragraph says that almost 203–268. [17] Gopal Prasad and Raghunathan M S 1983 On the con- all orbits are dense). A substantial progress on the gruence subgroup problem: Determination of the meta- conjecture was made by Dani [2] and later by Shah plectic kernel; Inv. Math. 71(1) 21–42. [24]. A major motivation for the conjecture was a [18] Raghunathan M S 1966 Vanishing theorems for coho- conjecture in number theory due to Oppenheim, mology groups associated to discrete subgroups of which says that the values of ‘irrational’ quadratic semi-simple Lie groups; Osaka J. Math. 3 243–256. forms on the integral lattice come arbitrarily close [19] Raghunathan M S 1967 Cohomology of arithmetic groups (I) and (II); Ann. Math. 86(2) 409–424; ibid to zero. This can be reformulated (the reformula- 87(2) 279–304. tion is due to Raghunathan) in terms of unipotent [20] Raghunathan M S 1976 On the congruence sub- orbits as described above. The Oppenheim conjec- group problem, Publications Mathematiques IHES; 46 ture was proved by Margulis. 107–161. The Dani–Raghunathan conjecture was proved [21] Raghunathan M S 1986 On the congruence subgroup 85(1) in full generality by Marina Ratner [23]. problem II; Inv. Math. 73–117. [22] Rajan C S 2004 Unique decomposition of tensor prod- ucts of irreducible representations of simple algebraic groups; Ann. Math. 160(2) 683–704. References [23] Ratner M 1991 On Raghunathan’s measure conjecture; Ann. Math. 134(3) 545–607. [1] Bhanu Murthy T S 1960 Plancherel’s measure for [24] Shah N A 1981 Uniformly distributed orbits of cer- the factor space SL(n, R)/SO(n); Sov. Math. Dokl. 1 tain flows on homogeneous spaces; Math. Ann. 289 860–862. 315–334. [2] Dani S G 1986 Orbits of horospherical flows; Duke [25] Shrawan Kumar 1988 Proof of the PRV conjecture; Math. J. 53(1) 177–188. Invent. Math. 93(1) 117–130. [3] Garland H and Raghunathan M S 1970 Fundamental [26] Sury B 1991 Congruence subgroup problem for domains for lattices in R-rank one semi-simple groups; anisotropic groups over semilocal rings; Proc. Indian Ann. Math. 92(2) 279–326. Acad. Sci. Math. Sci. 101(2) 87–110. [4] Harish-Chandra 1985 Collected works (New Delhi: [27] Venkataramana T N 1988 On super rigidity and arith- Narosa). meticity of lattices in semisimple groups over local [5] Knapp A and Zuckerman G 1982 Classification of fields of arbitrary characteristic; Invent. Math. 92(2) irreducible tempered representations of semi-simple 255–306.