Finite Hecke packets and infinite occupancy

Proof of Theorem 8.2

The idempotent relations in Standard fact 8.1 give the direct sum decomposition. Localization at \({\mathfrak m}\) kills every factor except the \({\mathfrak m}\) factor, where \(e_{{\mathfrak m}}\) becomes one. Thus the localization map identifies \(e_{{\mathfrak m}}V\) with \(V_{{\mathfrak m}}\). The final statement is the definition of support.

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Proof of Corollary 8.3

The maximal ideal of the Artin local algebra \({\mathbb T}_{{\mathfrak m}}\) is nilpotent. Hence a common power of every \(T_i-\lambda_i\) kills the local packet. On every other local factor, at least one \(T_i-\lambda_i\) is a unit, so no nonzero vector can satisfy all the displayed nilpotence conditions.

Conversely, suppose every \((T_i-\lambda_i)^N\) kills \(v\). Every monomial of total degree greater than \(d(N-1)\) contains one factor to exponent at least \(N\). Therefore a power of \({\mathfrak m}\) kills \(v\), which places \(v\) in the \({\mathfrak m}\) primary packet.

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Proof of Corollary 8.4

An idempotent linear map has minimal polynomial dividing \(X(X-1)\), which has distinct roots. It is diagonalizable with eigenvalues zero and one. Its trace and rank both equal the multiplicity of the eigenvalue one. A positive integer is nonzero in characteristic zero.

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Proof

The product decomposition gives the two central idempotents \((1,0)\) and \((0,1)\). Their ranges are the two local factors. The classes of \(u\) and \(v\) are nilpotent of orders two and three, respectively.

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Proof of Theorem 8.7

The displayed relations give

The nonnegative dimensions form a nonincreasing sequence of positive integers, so they eventually become constant. Once two consecutive dimensions agree, the corresponding inclusion is equality. Thus the ranges stabilize at a nonzero subspace, which is their intersection.

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Proof of Corollary 8.8

This is the contrapositive of Theorem 8.7.

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Proof of Lemma 8.10

Fix \(y\in Y_m\). For \(n\geq m+1\), let

Each \(A_n\) is a nonempty affine subspace of \(V_{m+1}\), and the sequence \((A_n)\) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in \(Y_{m+1}\) and maps to \(y\).

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Proof of Theorem 8.11

Choose \(0\neq y_1\in Y_1\). By Lemma 8.10, choose recursively \(y_{m+1}\in Y_{m+1}\) mapping to \(y_m\). Every \(y_m\) is nonzero because it maps to the nonzero vector \(y_1\). The sequence \((y_m)\) is a nonzero element of the inverse limit.

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Proof of Corollary 8.12

Surjectivity gives \(Y_m=V_m\). Apply Theorem 8.11.

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Proof

A vector in every tail range has its first \(m-1\) coordinates equal to zero for every \(m\). All of its coordinates vanish.

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