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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-8/6/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 6 / b / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The kernel and cokernel of an equivariant Fredholm operator are finite-dimensional representations. Define its equivariant Fredholm index in the representation ring of a compact group by

ind_{G} S = [ker S] - [coker S] .

For

G = S U (2)

write

H = ⨁_{n} E_{n} \otimes C^{m_{n}}

and

H^{'} = ⨁_{n} E_{n} \otimes C^{m_{n}^{'}}

, with finite multiplicities. By Schur lemma an intertwiner has blocks

I_{E_{n}} \otimes s_{n}

, where

s_{n} : C^{m_{n}} \to C^{m_{n}^{'}}

. If its rank is

r_{n}

, the kernel and cokernel multiplicities are

m_{n} - r_{n}

and

m_{n}^{'} - r_{n}

. Consequently

ind_{G} S = n \sum (m_{n} - m_{n}^{'}) [E_{n}] .

Only finitely many coefficients can be nonzero, because the global kernel and cokernel are finite-dimensional. Closed range is part of the Fredholm hypothesis and supplies the uniform lower bound on the remaining blocks; multiplicity matching alone would not establish that hypothesis. The ordinary index is obtained by taking dimensions,

ind S = \sum_{n} (n + 1) (m_{n} - m_{n}^{'})

. This calculation applies to any compact group with its finite-dimensional irreducibles in place of the

E_{n}

.

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