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Baruch's paper (Annals of Math. 158 (2003), 207-252) proves the following statement, originally claimed by Kirillov.

Theorem. Let $G=\mathrm{GL}_n(K)$ where $K$ is either $\mathbb{R}$ or $\mathbb{C}$, and let $P=P_n(K)$ be the subgroup of matrices in $\mathrm{GL}_n(K)$ consisting of matrices whose last row is $(0,0,\dots,0,1)$. If $\pi$ is an irreducible unitary representation of $G$ on a Hilbert space $H$ then $\pi|P$ is irreducible.

Representation theory is not my expertise, so the following question might be very basic. I might add further questions here if that is acceptable.

Question 1. On page 208 the author considers an arbitrary bounded linear operator $R:H\to H$ which commutes with $\pi(p)$ for all $p\in P$. He also considers, for any $f\in C_c^\infty(G)$, the distribution $\Lambda_R(f)=\mathrm{trace}(R\pi(f))$. Then he says: "It is easy to see that $\Lambda_R$ is an eigendistribution with respect to the center of the universal enveloping algebra associated to $G$." What is the meaning of "eigendistribution" here, and why is the statement true?

Baruch's paper (Annals of Math. 158 (2003), 207-252) proves the following statement, originally claimed by Kirillov.

Theorem. Let $G=\mathrm{GL}_n(K)$ where $K$ is either $\mathbb{R}$ or $\mathbb{C}$, and let $P=P_n(K)$ be the subgroup of matrices in $\mathrm{GL}_n(K)$ consisting of matrices whose last row is $(0,0,\dots,0,1)$. If $\pi$ is an irreducible unitary representation of $G$ on a Hilbert space $H$ then $\pi|P$ is irreducible.

Representation theory is not my expertise, so the following question might be very basic. I might add further questions here if that is acceptable.

Question 1. On page 208 the author considers an arbitrary bounded linear operator $R:H\to H$ which commutes with $\pi(p)$ for all $p\in P$. He also considers, for any $f\in C_c^\infty(G)$, the distribution $\Lambda_R(f)=\mathrm{trace}(R\pi(f))$. Then he says: "It is easy to see that $\Lambda_R$ is an eigendistribution with respect to the center of the universal enveloping algebra associated to $G$." What is the meaning of "eigendistribution" here, and why is the statement true?

Baruch's paper (Annals of Math. 158 (2003), 207-252) proves the following statement, originally claimed by Kirillov.

Theorem. Let $G=\mathrm{GL}_n(K)$ where $K$ is either $\mathbb{R}$ or $\mathbb{C}$, and let $P=P_n(K)$ be the subgroup of matrices in $\mathrm{GL}_n(K)$ consisting of matrices whose last row is $(0,0,\dots,0,1)$. If $\pi$ is an irreducible unitary representation of $G$ on a Hilbert space $H$ then $\pi|P$ is irreducible.

Representation theory is not my expertise, so the following question might be very basic.

Question. On page 208 the author considers an arbitrary bounded linear operator $R:H\to H$ which commutes with $\pi(p)$ for all $p\in P$. He also considers, for any $f\in C_c^\infty(G)$, the distribution $\Lambda_R(f)=\mathrm{trace}(R\pi(f))$. Then he says: "It is easy to see that $\Lambda_R$ is an eigendistribution with respect to the center of the universal enveloping algebra associated to $G$." What is the meaning of "eigendistribution" here, and why is the statement true?

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GH from MO
GH from MO
  • 105.3k
  • 8
  • 293
  • 398

Baruch's proof of Kirillov's conjecture

Baruch's paper (Annals of Math. 158 (2003), 207-252) proves the following statement, originally claimed by Kirillov.

Theorem. Let $G=\mathrm{GL}_n(K)$ where $K$ is either $\mathbb{R}$ or $\mathbb{C}$, and let $P=P_n(K)$ be the subgroup of matrices in $\mathrm{GL}_n(K)$ consisting of matrices whose last row is $(0,0,\dots,0,1)$. If $\pi$ is an irreducible unitary representation of $G$ on a Hilbert space $H$ then $\pi|P$ is irreducible.

Representation theory is not my expertise, so the following question might be very basic. I might add further questions here if that is acceptable.

Question 1. On page 208 the author considers an arbitrary bounded linear operator $R:H\to H$ which commutes with $\pi(p)$ for all $p\in P$. He also considers, for any $f\in C_c^\infty(G)$, the distribution $\Lambda_R(f)=\mathrm{trace}(R\pi(f))$. Then he says: "It is easy to see that $\Lambda_R$ is an eigendistribution with respect to the center of the universal enveloping algebra associated to $G$." What is the meaning of "eigendistribution" here, and why is the statement true?