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INVARIANT DIFFERENTIAL OPERATORS ON THE MINKOWSKI-EUCLID SPACE

  • Received : 2012.01.07
  • Published : 2013.03.01

Abstract

For two positive integers $m$ and $n$, let $\mathcal{P}_n$ be the open convex cone in $\mathbb{R}^{n(n+1)/2}$ consisting of positive definite $n{\times}n$ real symmetric matrices and let $\mathbb{R}^{(m,n)}$ be the set of all $m{\times}n$ real matrices. In this paper, we investigate differential operators on the non-reductive homogeneous space $\mathcal{P}_n{\times}\mathbb{R}^{(m,n)}$ that are invariant under the natural action of the semidirect product group $GL(n,\mathbb{R}){\times}\mathbb{R}^{(m,n)}$ on the Minkowski-Euclid space $\mathcal{P}_n{\times}\mathbb{R}^{(m,n)}$. These invariant differential operators play an important role in the theory of automorphic forms on $GL(n,\mathbb{R}){\times}\mathbb{R}^{(m,n)}$ generalizing that of automorphic forms on $GL(n,\mathbb{R})$.

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References

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