SL2(R) pp 191-203 | Cite as

Partial Differential Operators

  • Serge Lang
Part of the Graduate Texts in Mathematics book series (GTM, volume 105)


So far we have avoided to a large extent the more refined behavior of functions with respect to Lie derivatives. For the theory of spherical functions, we dealt with eigenvectors of convolution operators. The time has come to relate some invariants we have found in the representation theory with some of the invariant differential operators on G. Bargmann [Ba] saw how coefficient functions are eigenfunctions of such operators, Harish-Chandra got a complete insight into the situation by determining the center of the algebra of invariant differential operators, the centralizer of K in this algebra. Gelfand characterized spherical functions as eigenfunctions of this centralizer. In this chapter, we give Harish-Chandra’s result that there are no other spherical functions, besides those described in Chapter IV, on SL 2(R) where the proofs are short and easy.


Differential Operator Spherical Function Casimir Operator Partial Differential Operator Regularity Theorem 
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Copyright information

© Springer-Verlag New York Inc. 1985

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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