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The absolute continuity of convolutions of orbital measures in \(SO(2n+1)\)

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Let G be a compact Lie group of Lie type \(B_{n},\) such as \(SO(2n+1)\). We characterize the tuples \((x_{1},\ldots ,x_{L})\) of the elements \(x_{j}\in G\) which have the property that the product of their conjugacy classes has non-empty interior. Equivalently, the convolution product of the orbital measures supported on their conjugacy classes is absolutely continuous with respect to Haar measure. The characterization depends on the dimensions of the largest eigenspaces of each \(x_{j}\). Such a characterization was previously only known for the compact Lie groups of type \(A_{n}\).

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Correspondence to Kathryn E. Hare.

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Communicated by Andreas Cap.

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This research is supported in part by NSERC 2016-03719.

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Hare, K.E. The absolute continuity of convolutions of orbital measures in \(SO(2n+1)\). Monatsh Math 199, 275–299 (2022).

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