Anthony W. Knapp

Intertwining Operators for Semisimple Groups, II

The purpose of the present paper is to expand the use of intertwining operators for semisimple Lie groups. In an earlier form (see [20]), these operators were meromorphic continuations of integral operators and exhibited equivalences among nonunitary principal series representations, those induced from a minimal parabolic subgroup. We demonstrated that there was an intimate connection between these operators and the irreducibility of the principal series, on the one hand, and the unitarity of the analytically continued representations (the complementary series), on the other hand. In the intervening years we have generalized the setting for intertwining operators significantly, and we have learned of new applications. For the most part, the expansion in the setting is that minimal parabolic subgroups have now been replaced by arbitrary parabolic subgroups M A N , and the representations that are studied are induced from a representation of M A N that is irreducible unitary on M, is one-dimensional on A, and is trivial on N. The expanded theory allows us to determine the degree of reducibility of all series of representations appearing in the Plancherel formula of the group, and to study complementary series attached to them. Such intertwining operators have also now found major applications in classifying representations (see Langlands [29] and Knapp-Zuckerman [-25]) and appear to have significance for some problems in number theory. Our objective in this paper is threefold: to develop the analytic properties of intertwining operators in what seems to be the appropriate degree of generality, to give a dimension formula for the commuting algebras of the unitary representations induced when the representation of M is in the discrete series and the character of A is unitary (and to give some further insights into these representations), and to illustrate a technique for dealing with complementary series. To make it possible to be more specific, we introduce some notation. Let G be a reductive group whose identity component has compact center; the precise axioms for G are given in w 1. Fix a Cartan involution 0 for G, and let P be a parabolic subgroup of G. Then P c~ OP decomposes into the product of commuting subgroups M and A, where A is a vector group and M satisfies the

Singular Integrals and the Principal Series, II

A further study is made of the intertwining operators for the representations of the simple Lie groups of real rank one. We extend and apply this to the higher rank case, obtaining various results dealing with reducibility of principal series and existence of complementary series.

Correction and addition to Szegö kernels associated with discrete series

David Vogan has pointed out that Lemma 5.3 is incorrect, even for matrix groups, and therefore some changes are needed in the statements of the main theorems. The changes in question are not decisive, but we feel that the accurately stated versions of the theorems should be in the literature. Actually, when changes are needed, the new results yield more Szegi5 mappings than were originally predicted and in that sense represent an improvement of the original results. Vogan also suggested the statement below of Theorem A as an approach to making the necessary changes. To correct matters, delete Lemma 5.3 and introduce M I=Mo(Fc~T ), where F is the finite group defined in the proof of Lemma 5.3. Redefine oa on p. 176 to be the restriction of zx(M1) to the Ml-cyclic subspace H a generated by q~x. As in Proposition 5.5, we can conclude that cr~ is irreducible and has the stated highest weight and highest weight vector. The character ix gives the values of crx on elements of F ~ T , instead of F. For the most part, we can then replace subsequent occurrences of M by M 1 and of induction from MAN by induction from M1AN, and the results through the end of w 10 go through, with their new interpretations. (At the beginning of w delete the fourth paragraph and then define A(rr, v) directly in the obvious fashion.) No changes are needed in w167 11-12. Qualitatively the result is that the Szeg6 mapping f--*Sf now operates on a different domain of functions but otherwise has the same properties as in Theorem 1.1. The new domain is smooth functions from K into the redefined Hx that transform under the smaller group M1 according to the redefined aa. In representation-theoretic terms, the Szeg6 map S gives an intertwining operator between a representation W(a~, 2p + v) induced from MaAN to G (rather than MAN to G) and the discrete series ~a We can use this result to get an explicit quotient map to ~A from a representation induced from MAN to G. To this end, let

Structure Theory of Semisimple Groups

Every complex semisimple Lie algebra has a compact real form, as a consequence of a particular normalization of root vectors whose construction uses the Isomorphism Theorem of Chapter II. If go is a real semisimple Lie algebra, then the use of a compact real form of (g0)ℂ leads to the construction of a “Cartan involution” θ of go. This involution has the property that if go = t 0 ⊕ p0 is the corresponding eigenspace decomposition or “Cartan decomposition”, then to ⊕ ipo is a compact real form of (g0)ℂ. Any two Cartan involutions of go are conjugate by an inner automorphism. The Cartan decomposition generalizes the decomposition of a classical matrix Lie algebra into its skew-Hermitian and Hermitian parts.

Finite-Dimensional Representations

In any finite-dimensional representation of a complex semisimple Lie algebra g, a Cartan subalgebra h acts completely reducibly, the simultaneous eigenvalues being called “weights.” Once a positive system for the roots Δ+(g, h) has been fixed, one can speak of highest weights. The Theorem of the Highest Weight says that irreducible finite-dimensional representations are characterized by their highest weights and that the highest weight can be any dominant algebraically integral linear functional on h. The hard step in the proof is the construction of an irreducible representation corresponding to a given dominant algebraically integral form. This step is carried out by using “Verma modules,” which are universal highest weight modules.

Equivariant maps onto minimal flows

Let G be a connected Lie group. (X, G) is aflow i fX is a compact Hausdorff space with a joint ly continuous group action by G. T h e flow (X, G) is minimal if every orbit is dense or, equivalently, if X has no proper closed non-empty invariant set. We wish to capture two ideas about minimal sets. The first is that if (X, G) is a minimal flow, then not only is each orbit dense, but also each orbit actually winds a round X in much the same way that a Kronecker line on the torus winds a round the torus. This fact is made precise and explained fur ther in T h e o r e m 2.1 and the discussion immediately following it. As a consequence of this theorem, we obtain as Theo rem 2.2 a s tatement about equivariant maps of flows onto minimal flows, which generalizes the main result of Chu and Geraghty in [2]. The second idea is that the space X of a minimal flow should have some homogenei ty property. A known result, due to A. A. Markoff [7], is that if X is finite-dimensional, then X has the same dimension at each point. The conjecture that X has a transitive set of homeomorphisms commut ing with G is shown to be false by enlarging the space of Floyds example [6] and making it into a flow unde r the reals in the usual way. Instead, our result is of a relative ra ther than an absolute nature. Namely, if rr is an equivariant mapping between minimal flows (X, G) and (Y, G), then under suitable conditions X is the bundle space of a fiber bundle with base space Y and with projection zr. Such a result is proved as Theo rem 3.1 under the assumption that everything is differentiable.

Wedderburn–Artin Ring Theory

This chapter studies finite-dimensional associative division algebras, as well as other finite-dimensional associative algebras and closely related rings. The chapter is in two parts that overlap slightly in Section 6. The first part gives the structure theory of the rings in question, and the second part aims at understanding limitations imposed by the structure of a division ring.

Infinite Field Extensions

This chapter provides algebraic background for directly addressing some simple-sounding yet fundamental questions in algebraic geometry. All the questions relate to the set of simultaneous zeros of finitely many polynomials in n variables over a field.

Methods of Algebraic Geometry

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field.

The Number Theory of Algebraic Curves

This chapter investigates algebraic curves from the point of view of their function fields, using methods analogous to those used in studying algebraic number fields.