K-invariants in the algebra $U(g) \otimes C(p)$ for the group SU(2, 1) (CROSBI ID 217031)
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Prlić, Ana
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K-invariants in the algebra $U(g) \otimes C(p)$ for the group SU(2, 1)
Let $g = k \oplus p$ be the Cartan decomposition of the complexified Lie algebra g = sl(3, C) of the group G = SU(2, 1). Let K = S(U (2) × U (1)) ; so K is a maximal compact subgroup of G. Let U(g) be the universal enveloping algebra of g, and let C(p) be the Clifford algebra with respect to the trace form B(X, Y ) = tr(XY ) on p. We are going to prove that the algebra of K–invariants in U (g) \otimes C(p) is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for (g, K)-modules. Along the way we will also recover the (well known) structure of the algebra U (g)^K.
Lie group ; Lie algebra ; representation ; special unipotent rep- resentation ; Dirac operator ; Dirac cohomology
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