Lie groups and algebraic groups M S RAGHUNATHAN and T N VENKATARAMANA∗ School of Mathematics, 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 mathematicians 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 group simple if G has no connected is the study of symmetries of any mathemati- normal closed subgroups 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 class of groups are simple algebraic equations. groups. They can be defined over an arbitrary field. The most ubiquitous class of groups 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. Representation theory 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 Lie group 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 subgroup 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 unitary group 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 Lie algebra is a direct sum of irreducible result in algebraic topology; this is an instance (finite dimensional) representations of G each of of geometric properties of G yielding algebraic which has dimension 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 mathematician 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 circle group), 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-simple Lie group G has discrete series if translations. One has the fundamental theorem of and only if G has a connected maximal abelian Hermann Weyl, Armand Borel 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 symmetric space 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).