An Introduction to Harmonic Analysis on Semisimple Lie by V. S. Varadarajan

By V. S. Varadarajan

Now in paperback, this graduate-level textbook is a superb creation to the illustration conception of semi-simple Lie teams. Professor Varadarajan emphasizes the improvement of significant topics within the context of specified examples. He starts off with an account of compact teams and discusses the Harish-Chandra modules of SL(2,R) and SL(2,C). next chapters introduce the Plancherel formulation and Schwartz areas, and exhibit how those bring about the Harish-Chandra thought of Eisenstein integrals. the ultimate sections think about the irreducible characters of semi-simple Lie teams, and comprise particular calculations of SL(2,R). The e-book concludes with appendices sketching a few easy subject matters and with a entire consultant to additional examining. This marvelous quantity is very compatible for college students in algebra and research, and for mathematicians requiring a readable account of the subject.

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Extra resources for An Introduction to Harmonic Analysis on Semisimple Lie Groups (Cambridge Studies in Advanced Mathematics)

Example text

Indeed, if 111 1, m 2 EM. 3), whence Thus D is a group. 3), that Hence H centralizes D. Finally, let k E K and m E M. 4), we have k ~ I «(1110)111 ~ l)k = (1I10)k(lII- l)k = (mk)O(nl) - I E D. Hence D is invariant under K as well as H and so D is normal in G. Observe next that as H n K = I and Mc:;: H, (1I10)m- 1 EH only if 1110 = 1. Since 0 is an isomorphism. 111 = 1 and so H n D = 1. Similarly, (1110)111- 1 E K implies 111 = 1 and so also K n D = I. Hence if we set G = G/ D and let H, K M be the respective images of K, K, M in G, it it follows that R is isomorphic to Hand R to K.

Then it is immediate that h fixes the coset Hk as well as the coset H. But by definition of a Frobenius group, only the identity fixes more than one letter. Thus k h of k for any h in H # and k in K #. In particular, (iii) holds. Moreover, for a fixed k of 1, the set r k = {k h I h E H} must consist of m = IHI distinct elements of K. But clearly for k, k' in K #, we have either r k = r k , or r k (J r k , = 0, the empty set. Hence IK# 1is a multiple of m and (ii) follows. Finally, by (iii), no element of K # centralizes any element of a conjugate of H #.

Moreover, G = HR with H <:l G. By definition of D, m, and m8 lie in the same coset of D for each 1/1 in M and consequently NI c:;: H n K. On the other hand, IDI = IMI, whence IG/ DI = IH IIKI/IMI = IH I KI/IM I. forcing IM I = IH n RI. \' denotes the image in G of the element x of G, we have jji< = (h k) = (hktjl) for all h in H, k in K. Thus G exists with the required properties. As usual, we identify H. K, M with their isomorphic images in G and drop the superscript on G. Thus G = HK, H <:J G, M = H n K, and for h in H, k in K.

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