Polynomials and Representations XXXII

We attempt to identify the irreducible rational representations of G = GL_n\mathbb C. From the last article, we may tensor it with a suitable power of det and assume it is polynomial.

One key ingredient is the following rather ambiguous statement.

  • Peter-Weyl Principle: any irrep can be embedded inside the ring of functions on G.

To make sense of this statement, we need to define a group action on the latter. This article is rather general; throughout, G is allowed to be any topological group and all representations are assumed to be continuous and finite-dimensional.

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C(G) as a Representation

Let C(G) be the ring of continuous functions G\to \mathbb{C}.

Definition. The action of G\times G on C(G) is defined as follows:

(x, y)\in G\times G, f \in C(G)\ \implies\ (x,y)\cdot f : G\to\mathbb{C}, g\mapsto f(x^{-1}gy).

One checks that (x',y')\cdot((x,y)\cdot f) = (x'x, y'y)\cdot f.

Some examples of functions in C(G) are the matrix coefficients; recall that these are functions of the form f : G \to \mathbb{C}, g\mapsto \alpha (\rho(v)) for some representation \rho : G\to GL(V), v\in V and \alpha \in V^\vee. A more economical way of writing this is f = \beta\circ \rho for some \beta \in \text{End}_{\mathbb C}(V)^\vee.

Proposition. Let \mathcal{O}(G) be the set of matrix coefficients. Then \mathcal{O}(G) is a (unital) \mathbb{C}-subalgebra of C(G).

Proof

To prove 1\in\mathcal{O}(G), let \rho :G \to \mathbb{C}^* be the trivial representation.

Closure under addition and multiplication: suppose \rho_i:G\to GL(V_i), \beta_i\in \text{End}(V_i)^\vee for i = 1,2; let f_i = \beta_i \circ \rho_i:G\to\mathbb{C} be the matrix coefficient. Taking:

\begin{aligned} \rho_1 \oplus \rho_2 : G \to GL(V_1 \oplus V_2), \ \ &\beta_1 \oplus \beta_2 \in \text{End}(V_1)^\vee \oplus \text{End}(V_2)^\vee \subseteq \text{End}(V_1 \oplus V_2)^\vee, \\ \rho_1\otimes \rho_2 : G\to GL(V_1 \otimes V_2), \ \ &\beta_1 \otimes \beta_2 \in \text{End}(V_1)^\vee \otimes \text{End}(V_2)^\vee \cong \text{End}(V_1 \otimes V_2)^\vee,\end{aligned}

we have f_1 + f_2 = (\beta_1 \oplus \beta_2) \circ (\rho_1 \oplus \rho_2) and f_1 f_2 = (\beta_1 \otimes \beta_2) \circ(\rho_1 \otimes \rho_2). ♦

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Identifying Matrix Coefficients in C(G)

Next we have:

Lemma. For any f\in C(G), the following are equivalent.

  1. The span of all (g,g')\cdot f over all (g, g')\in G \times G is finite-dimensional.
  2. The span of all (g,1)\cdot f over all g\in G is finite-dimensional.
  3. The span of all (1,g)\cdot f over all g\in G is finite-dimensional.
  4. f is a matrix coefficient.

Proof

That 1 ⇒ 2, 3 is obvious. Let us now prove 4 ⇒  1. Suppose f = \beta\circ \rho for \rho : G\to GL(V) and \beta : \text{End}(V) \to \mathbb{C}. Then the function (x, y)f : G\to \mathbb{C} takes

g \mapsto f(x^{-1}gy) = \beta(\rho(x^{-1}) \rho(g) \rho(y)) = \beta'(\rho(g))

where \beta' : \text{End} (V) \to \mathbb{C} is a linear map which takes \phi \mapsto \beta(\rho (x^{-1})\phi\rho(y)). Hence the span of all (x,y)f lies in \{\beta' \circ \rho : \beta' \in \text{End}(V)^\vee\} and (1) follows since \text{End}(V)^\vee is finite-dimensional.

Finally, we prove 3 ⇒ 4; the case of 2 ⇒ 4 is similar. Let V be the space spanned by all (1, g)f, so V is a finite-dimensional subspace of C(G) which is (1\times G)-invariant. Let

  • \rho : G \to GL(V) be given by the action of (1\times G) on V;
  • \beta : \text{End}(V) \to \mathbb{C} be the map \alpha \mapsto (\alpha(f))(e_G).

Now tracing through \beta\circ \rho : G \to \mathbb{C} gives:

\overbrace{g}^{\in G}\ \mapsto \overbrace{(f_1\in V \mapsto (x\mapsto f_1(xg)))}^{\in GL(V)} \mapsto \left.(x\mapsto f(xg))\right|_{x=e} = f(g).

Thus f = \beta\circ \rho is a matrix coefficient. [Note: for 2 ⇒ 4, one needs to take the dual of V.]   ♦

Corollary. \mathcal{O}(G) is the sum of all finite-dimensional G\times G-subrepresentations of C(G).

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Weak Peter-Weyl Theorem

Let V be a representation of G; the vector space End(V) becomes a G\times G-rep via:

(x,y) : (\alpha : V\to V) \mapsto (\rho_V(x)\circ \alpha \circ\rho_V(y^{-1}) : V\to V).

Another way of seeing this is: \text{End}(V) \cong V\otimes V^\vee naturally and the action of G\times G on the LHS is obtained from the RHS. Next, we obtain a linear map:

\text{End}(V)^\vee\to \mathcal{O}(G),\qquad \beta \mapsto (\beta \circ \rho_V : G\to \mathbb{C}). (*)

This is G\times G-equivariant:

  • Indeed (xy) takes \alpha\in \text{End}(V) to \rho_V(x) \alpha \rho_V(y^{-1}) so dually \beta \in \text{End}(V)^\vee is taken to the function (\alpha \in \text{End}(V)) \mapsto \beta(\rho_V(x^{-1})\alpha \rho_V(y)).
  • In \mathcal{O}(G), (xy) takes f:G\to\mathbb{C} to g\mapsto f(x^{-1}gy). Letting f = \beta\circ \rho_V it gets taken to g \mapsto \beta(\rho_V(x^{-1}) \rho_V(g) \rho_V(y)) so the following commutes:

\begin{matrix} \text{End}(V)^\vee &\longrightarrow & \mathcal{O}(G)\\ \downarrow & &\downarrow & \text{action of }(x,y)\\ \text{End}(V)^\vee &\longrightarrow &\mathcal{O}(G)\end{matrix}

warningWe have \text{End}(V) \cong V \otimes V^\vee so its dual V^\vee \otimes V is naturally isomorphic to End(V) as vector spaces and as G-reps, but not as G\times G-reps. This is because the isomorphism V^\vee \otimes V \cong V\otimes V^\vee requires a twist of the two components, which is not G\times G-equivariant.

Now suppose G is compact Hausdorff.

Lemma. Let V, W be irreducible representations of G. Then:

  • \text{End}(V) is an irreducible representation of G\times G;
  • \text{End}(V) \cong \text{End}(W) as G\times G-reps if and only if V\cong W as G-reps.

Proof

The character of End(V) is: \chi_{\text{End}(V)}(g_1, g_2) = \chi_{V}(g_1) \chi_{V^\vee}(g_2). Taking the inner product:

\begin{aligned}\int_{G\times G} \chi_{\text{End}(V)}(g_1, g_2) \chi_{\text{End}(W)}(g_1, g_2) d(g_1, g_2) &=\int_G \int_G \chi_{V}(g_1) \chi_{V^\vee}(g_2)  \chi_{W}(g_1) \chi_{W^\vee}(g_2) dg_1 \, dg_2\\ &= \left(\int_G \chi_{V}(g_1) \chi_{W}(g_1) dg_1\right) \left(\int_G \chi_{V^\vee}(g_2) \chi_{W^\vee}(g_2) dg_2\right)\\&= \begin{cases} 1, \ \ &\text{if } V\cong W; \\ 0,\ \ &\text{otherwise.}\end{cases}\end{aligned}

This proves both claims together. ♦

Taking the direct sum of (*) over all irreps V of G, we obtain:

\displaystyle\bigoplus_{V \text{irrep of } G} \text{End}(V)^\vee\to \mathcal{O}(G).

Weak Peter-Weyl Theorem. The above map is an isomorphism.

Proof

To show injectivity: the kernel of the map is a G\times G-submodule, so it is a direct sum of G\times G-irreps \text{End}(V)^\vee. On the other hand, \text{End}(V)^\vee \to \mathcal{O}(G) clearly cannot map to 0, since that would mean \rho_V = 0.

To show surjectivity: each f\in \mathcal{O}(G) is, by definition, of the form \beta \circ \rho for some \rho : G \to GL(V), \beta \in \text{End}(V)^\vee. Thus lies in the image. ♦

We remind the reader that this theorem only works when G is compact Hausdorff.

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