# D-modules, representation theory, and quantum groups: by Louis Boutet de Monvel, C. De Concini, C. Procesi

By Louis Boutet de Monvel, C. De Concini, C. Procesi

Lectures Given on the 2d consultation of the Centro Internazionale Matematico Estivo (C.I.M.E.) Held in Venezia, Italy, June 12-20, 1992

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2, 1-16; Funct. Anal. appl. 5 (1971), 89-101. BjSrk J. , Rings of Differential Operators, North Holland, 1979. Borel A. et al. [1], Intersection cohomology, vol. , Birkhs 1984. - - [2], Algebraic D-modules, vol. 2, Perspect. , Academic Press, 1987. 29 Boutet de Monvel L. [1], On the index of Toeplitz operators of several complex variables, Inventiones Math. 50 (1979), 249-272; S6minaire EDP 1979, Ecole Polytechnique. - - [2], Syst~mes presqu'elliptiques: une autre demonstration de la formule de l'indice, Ast~risque 131 (1985), 201-216.

Bernstein I. , Modules over rings of differential operators. An investigation of the fundamental solution of equations with constant coefficients, Funkc. Anal i Prilozen 5 (1971), no. 2, 1-16; Funct. Anal. appl. 5 (1971), 89-101. BjSrk J. , Rings of Differential Operators, North Holland, 1979. Borel A. et al. [1], Intersection cohomology, vol. , Birkhs 1984. - - [2], Algebraic D-modules, vol. 2, Perspect. , Academic Press, 1987. 29 Boutet de Monvel L. [1], On the index of Toeplitz operators of several complex variables, Inventiones Math.

Multiplication and comultiplication are exchanged by duality. For instance the comultiplication in the algebra of functions on a group gives by duality the convolution. We want to point out a useful construction. Given an algebra A, it can be considered as an A | A ~ module setting: ( a | b)c := acb. 1) A ~ A| l| , A| A0 is clearly a homomorphism and so it induces a module structure of A on itself. This action is called the Adjoint action and will be indicated by ad(a)(b) := Z i uibS(vi), if : A(a) = Z ui | vi.