Irreducibility of discrete series representations for semisimple symmetric spaces.

*(English)*Zbl 0733.22008
Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14, 191-221 (1988).

[For the entire collection see Zbl 0694.00014.]

Let G be a connected reductive Lie group in Harish Chandra’s class. Let \(\sigma\) be an involution of G and H a subgroup between the fixed point set of \(\sigma\) and its identity component. The representations of G on irreducible subrepresentations of G on \(L^ 2(G/H)\) are called the discrete series representations of G on the symmetric space G/H.

After a significant work of M. Flensted-Jensen [Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006)], T. Oshima and T. Matsuki [Adv. Stud. Pure Math. 4, 331-390 (1984; Zbl 0577.22012)] gave a detailed description of all discrete series representations on G/H. For each X in a certain parameter set P, they constructed a unitary representation A(X) and an embedding of A(X) in \(L^ 2(G/H)_ d\), the sum of all discrete series on G/H. Then they proved that \(L^ 2(G/H)_ d=\oplus_{A\in P}A(X).\)

The author considers the irreducibility question and proves that the representations A(X) are irreducible or zero. The proof is as follows. The result was established by Oshima-Matsuki (loc. cit.) when X is generic. For the general case the author proceeds by reduction to the generic case, the method being based on the “translation principle” of Jantzen and Zuckerman. He shows that any A(X) can be obtained from some generic Y by tensoring A(Y) with an appropriate finite dimensional representation of G. Then he studies when such a construction preserves irreducibility.

Let G be a connected reductive Lie group in Harish Chandra’s class. Let \(\sigma\) be an involution of G and H a subgroup between the fixed point set of \(\sigma\) and its identity component. The representations of G on irreducible subrepresentations of G on \(L^ 2(G/H)\) are called the discrete series representations of G on the symmetric space G/H.

After a significant work of M. Flensted-Jensen [Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006)], T. Oshima and T. Matsuki [Adv. Stud. Pure Math. 4, 331-390 (1984; Zbl 0577.22012)] gave a detailed description of all discrete series representations on G/H. For each X in a certain parameter set P, they constructed a unitary representation A(X) and an embedding of A(X) in \(L^ 2(G/H)_ d\), the sum of all discrete series on G/H. Then they proved that \(L^ 2(G/H)_ d=\oplus_{A\in P}A(X).\)

The author considers the irreducibility question and proves that the representations A(X) are irreducible or zero. The proof is as follows. The result was established by Oshima-Matsuki (loc. cit.) when X is generic. For the general case the author proceeds by reduction to the generic case, the method being based on the “translation principle” of Jantzen and Zuckerman. He shows that any A(X) can be obtained from some generic Y by tensoring A(Y) with an appropriate finite dimensional representation of G. Then he studies when such a construction preserves irreducibility.

Reviewer: H.Fujiwara (Iizuka)

##### MSC:

22E46 | Semisimple Lie groups and their representations |

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

43A85 | Harmonic analysis on homogeneous spaces |

53C35 | Differential geometry of symmetric spaces |