These lecture notes are devoted to spherical functions on Olshanski spherical pairs and generalizations. In the first chapter, results on Olshanski spherical pairs from [G. I. Ol’shanskij, Adv. Stud. Contemp. Math. 7, 269-463 (1990; Zbl 0724.22020)] are reviewed; by definition, an Olshanski spherical pair \((G,K)\) is the inductive limit of an increasing sequence of Gelfand pairs. As in the finite dimensional case, it turns out that a function in \(P_1(K\backslash G/K)\) is spherical iff the representation associated to it by the Gelfand-Naimark-Segal construction is irreducible.
The second chapter deals with classical and shifted Schur functions, and their relation to elementary symmetric functions [see A.Yu. Okun’kov and G. Ol’shanskij, St. Petersbg. Math. J. 9, No. 2, 239–300 (1998; Zbl 0894.05053)]. Then, two main examples are treated:
Chapter 3 discusses the space of infinite dimensional Hermitian matrices with the action of the infinite dimensional unitary group. It is shown that spherical functions are then exactly Fourier transforms of orbital measures, and their asymptotic behaviour is described. These results are from [G. Ol’shanskij and A.M. Vershik, Transl., Ser. 2, Am. Math. Soc. 175(31), 137–175 (1996; Zbl 0853.22016)]. The second main example (Chapter 4) is the infinite dimensional unitary group, viewed as a symmetric space with the conjugation action. In the finite dimensional case, spherical functions can then be expressed through characters of irreducible representations; as the dimension tends to infinity, they are written by means of Voiculescu functions and their asymptotics are discussed [see A. M. Vershik and S. V. Kerov, Funct. Anal. Appl. 15, 246–255 (1982; Zbl 0507.20006) for the results and A. Okounkov and G. Olshanski, Transl., Ser. 2, Am. Math. Soc. 181(35), 245–271 (1998; Zbl 0941.17008) for the proofs]. The last chapter is a very short review of the analogous results for inductive limits of compact symmetric spaces.
The text is well-written, though clearly recognizable as lecture notes. It is well-suited for a first introduction to the field.
Contents: 1. Olshanski spherical pairs; 2. Schur functions; 3. Infinite dimensional Hermitian matrices; 4. Infinite dimensional unitary group; 5. Inductive limits of compact symmetric spaces.