We study some extension of a discrete Heisenberg group coming from the theory of loop-groups and find invariants of conjugacy classes in this group. In some cases including the case of the integer Heisenberg group we make these invariants more explicit.
Conjugacy classes of involutions and Kazhdan-Lusztig cells
Quotients for sheets of conjugacy classes
Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument
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