Spherical Inversion on SLn(R)

1. Front Matter

   Pages i-xx

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2. Iwasawa Decomposition and Positivity

   • Jay Jorgenson, Serge Lang

   Pages 1-32

3. Invariant Differential Operators and the Iwasawa Direct Image

   • Jay Jorgenson, Serge Lang

   Pages 33-73

4. Characters, Eigenfunctions, Spherical Kernel and W-Invariance

   • Jay Jorgenson, Serge Lang

   Pages 75-129

5. Convolutions, Spherical Functions and the Mellin Transform

   • Jay Jorgenson, Serge Lang

   Pages 131-175

6. Gelfand-Naimark Decomposition and the Harish-Chandra c-Function

   • Jay Jorgenson, Serge Lang

   Pages 177-218

7. Polar Decomposition

   • Jay Jorgenson, Serge Lang

   Pages 219-254

8. The Casimir Operator

   • Jay Jorgenson, Serge Lang

   Pages 255-275

9. The Harish-Chandra Series and Spherical Inversion

   • Jay Jorgenson, Serge Lang

   Pages 277-308

10. General Inversion Theorems

    • Jay Jorgenson, Serge Lang

    Pages 309-324

11. The Harish-Chandra Schwartz Space (HCS) and Anker’s Proof of Inversion

    • Jay Jorgenson, Serge Lang

    Pages 325-371

12. Tube Domains and the L ¹ (Even L ^p ) HCS Spaces

    • Jay Jorgenson, Serge Lang

    Pages 373-386

13. SL _n (C)

    • Jay Jorgenson, Serge Lang

    Pages 387-410

14. Back Matter

    Pages 411-426

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Harish-Chandra's general Plancherel inversion theorem admits a much shorter presentation for spherical functions. The authors have taken into account contributions by Helgason, Gangolli, Rosenberg, and Anker from the mid-1960s to 1990. Anker's simplification of spherical inversion on the Harish-Chandra Schwartz space had not yet made it into a book exposition. Previous expositions have dealt with a general, wide class of Lie groups. This has made access to the subject difficult for outsiders, who may wish to connect some aspects with several if not all other parts of mathematics, and do so in specific cases of intrinsic interest. The essential features of Harish-Chandra theory are exhibited on SLn(R), but hundreds of pages of background can be replaced by short direct verifications. The material becomes accessible to graduate students with especially no background in Lie groups and representation theory. Spherical inversion is sufficient to deal with the heat kernel, which is at the center of the authors' current research. The book will serve as a self-contained background for parts of this research.

• Convexity
• algebra
• differential operator
• integral
• integration
• lie group
• metric space
• operator
• proof
• representation theory
• transform theory

From the reviews:

"[This] book presents the essential features of the theory on SLn(R). This makes the book accessible to a wide class of readers, including nonexperts of Lie groups and representation theory and outsiders who would like to see connections of some aspects with other parts of mathematics. This feature is widely to be appreciated, together with the clearness of exposition and the way the book is structured." -Sergio Console, Zentralblatt

"This book is devoted to Harish-Chandra’s Plancherel inversion formula in the special case of the group SL_n(R) and for spherical functions. ... the book is easily accessible and essentially self contained." (A. Cap, Monatshefte für Mathematik, Vol. 140 (2), 2003)

"Roughly, this book offers a ‘functorial exposition’ of the theory of spherical functions developed in the late 1950s by Harish-Chandra, who never used the word ‘functor’. More seriously, the authors make a considerable effort to communicate the theory to ‘an outsider’. .... However, even an expert will notice several new and pleasing results like the smooth version of the Chevally restriction theorem in Chapter 1." (Tomasz Przebinda, Mathematical Reviews, Issue 2002 j)

"This excellent book is an original presentation of Harish-Chandra’s general results ... . Unlike previous expositions which dealt with general Lie groups, the present book presents the essential features of the theory on SL_n(R). This makes the book accessible to a wide class of readers, including nonexperts ... . This feature is widely to be appreciated, together with the clearness of exposition and the way the book is structured. Very nice is, for instance, the ... table of the decompositions of Lie groups." (Sergio Console, Zentralblatt MATH, Vol. 973, 2001)

• Department of Mathematics, City College of New York, CUNY, New York, USA

  Jay Jorgenson

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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