Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-Chandra modules. On $\widehat G$ we naturally have the Fell topology and on $\widehat G_{adm}$ we can install a topology using the Langlands classification. There is a third possibility as follows: Let $\mathbb D$ denote the algebra of bi-invariant differential operators, which is isomorphic to $\mathbb{C}[X_1,\dots, X_n]$ for some $n$. On each $\pi\in\widehat G_{adm}$ the operators in $\mathbb D$ acts by scalars, so $\pi$ detyermines a point $\chi_\pi\in\mathbb{C}^n$, whose coordinates are the eigenvalues of the generators of $\mathbb D$. Further fix a maximal compact subgroup $K$. Then each $\tau\in\widehat K$ occurs with a finite multiplicity in $\pi|_K$, so $\pi$ defines an element of the discrete set $D=\prod_{\tau\in\widehat K}\mathbb{N}_0$. Together we get a map $$ P:\widehat G_{adm}\to\mathbb{C}^n\ \times\ D, $$ which by pullback also puts a topology on $\widehat{G}_{adm}$. My questions are:

1. Is $P$ injective?
2. How do the three topologies on $\widehat G$ compare? Which is finer and how large are the differences?
3. Do they all give the same Borel structure?

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-Chandra modules. Let $\mathbb D$ denote the algebra of bi-invariant differential operators and let $\mathbb{D}^*=\mathrm{Hom}_{alg}(\mathbb{D},\mathbb{C})$ which is isomorphic to $\mathbb{C}[X_1,\dots, X_n]$ for some $n$. On each $\pi\in\widehat G_{adm}$ the operators in $\mathbb D$ acts by scalars, so $\pi$ determines a point $\chi_\pi\in\mathbb{C}^n$, whose coordinates are the eigenvalues of the generators of $\mathbb D$. Further fix a maximal compact subgroup $K$. Then each $\tau\in\widehat K$ occurs with a finite multiplicity in $\pi|_K$, so for any subset $F\subset\widehat K$, the representation $\pi$ defines an element of the discrete set $D_F=\prod_{\tau\in F}\mathbb{N}_0$. Together we get a map $$ P_F:\widehat G_{adm}\to\mathbb{C}^n\ \times\ D_F. $$ My questions are:

1. If $F=\widehat K$, is $P$ injective?
2. For a given bounded set $B\subset \mathbb{C}^n$, is it true that there exists a finite set $F\subset\widehat K$ such that $P_F$ is injective on $P_\emptyset^{-1}(B)$?

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-Chandra modules. On $\widehat G$ we naturally have the Fell topology and on $\widehat G_{adm}$ we can install a topology using the Langlands classification. There is a third possibility as follows: Let $\mathbb D$ denote the algebra of bi-invariant differential operators, which is isomorphic to $\mathbb{C}[X_1,\dots, X_n]$ for some $n$. On each $\pi\in\widehat G_{adm}$ the operators in $\mathbb D$ acts by scalars, so $\pi$ detyermines a point $\chi_\pi\in\mathbb{C}^n$, whose coordinates are the eigenvalues of the generators of $\mathbb D$. Further fix a maximal compact subgroup $K$. Then each $\tau\in\widehat K$ occurs with a finite multiplicity in $\pi|_K$, so $\pi$ defines an element of the discrete set $D=\prod_{\tau\in\widehat K}\mathbb{N}_0$. Together we get a map $$ P:\widehat G_{adm}\to\mathbb{C}^n\ \times\ D, $$ which by pullback also intsalls a topology on $\widehat{G}_{adm}$. My questions are:

1. Is $P$ injective?
2. How do the three topologies on $\widehat G$ compare? Which is finer and how large are the differences?
3. Do they all give the same Borel structure?

Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-Chandra modules. On $\widehat G$ we naturally have the Fell topology and on $\widehat G_{adm}$ we can install a topology using the Langlands classification. There is a third possibility as follows: Let $\mathbb D$ denote the algebra of bi-invariant differential operators, which is isomorphic to $\mathbb{C}[X_1,\dots, X_n]$ for some $n$. On each $\pi\in\widehat G_{adm}$ the operators in $\mathbb D$ acts by scalars, so $\pi$ detyermines a point $\chi_\pi\in\mathbb{C}^n$, whose coordinates are the eigenvalues of the generators of $\mathbb D$. Further fix a maximal compact subgroup $K$. Then each $\tau\in\widehat K$ occurs with a finite multiplicity in $\pi|_K$, so $\pi$ defines an element of the discrete set $D=\prod_{\tau\in\widehat K}\mathbb{N}_0$. Together we get a map $$ P:\widehat G_{adm}\to\mathbb{C}^n\ \times\ D, $$ which by pullback also puts a topology on $\widehat{G}_{adm}$. My questions are:

1. Is $P$ injective?
2. How do the three topologies on $\widehat G$ compare? Which is finer and how large are the differences?
3. Do they all give the same Borel structure?