Manuscript Oai069-A@1 · Oai069-A.tex · lemma · a4913aab

Lemma: Orthogonal character projectors

An explicit abelian local-system category › Rank-one local systems on the elliptic curve · lines 443–468 · proof included or linked

Oai069-A@1#lem:projectors

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For unitary $U$ with $U^{d} = I$ , set

P_{k} = \frac{1}{d} a = 0 \sum^{d - 1} ω^{- k a} U^{a} .

Then

P_{k}^{*} = P_{k}, P_{k} P_{l} = δ_{k l} P_{k}, k \sum P_{k} = I, U P_{k} = ω^{k} P_{k} .

Proof. For every $b \in Z / d Z$ ,

a = 0 \sum^{d - 1} ω^{ab} = {d, 0, b = 0, b \neq = 0.

Expanding the products of the finite sums proves orthogonality, the sum identity, and the eigenvalue identity. Replacing $a$ by $- a$ , and using $U^{*} = U^{- 1}$ , proves self-adjointness.

Displayed formulas in this passage

unlabeled · line 446
$P_{k} = \frac{1}{d} a = 0 \sum^{d - 1} ω^{- k a} U^{a} .$
unlabeled · line 450
$P_{k}^{*} = P_{k}, P_{k} P_{l} = δ_{k l} P_{k}, k \sum P_{k} = I, U P_{k} = ω^{k} P_{k} .$
unlabeled · line 460
$a = 0 \sum^{d - 1} ω^{ab} = {d, 0, b = 0, b \neq = 0.$

Lemma: Orthogonal character projectors

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