Admissible representations of GL(3) The finite dimensional representation of $GL(2,\mathbb{R})$ are obtained by tensoring the symmetric powers of standard representations of $GL(2,\mathbb{R})$ with the character $\chi \circ \textrm{det}$ where $\chi: \mathbb{R}^\times \to \mathbb{C}^\times$.}$
Is there a similar description for finite dimensional representations of $GL(3)$? For instance tensoring with the $\Gamma_{a,b}$ with the character $\chi \circ \det$ where $\Gamma_{a,b}$ is an irreducible rep. of $GL(3,\mathbb{R})$ with highest weight $aL_1 - b L_3$? Here $L_i : T(\mathbb{R}) \to \mathbb{C}$ is the linear functional taking the diagonal matrices  $\begin{bmatrix} a_1 & & \\ & a_2 & \\  & & a_3 \end{bmatrix}$ of the standard maximal torus $T(\mathbb{R})$  to $a_i$. 
 A: Yes. Let $G$ be any connected reductive $\mathbf{R}$-group whose derived group $G'$ is simply connected in the sense of connected semisimple algebraic groups (e.g., $G = {\rm{GL}}_n$, so $G' = {\rm{SL}}_n$). It seems the question concerns relating (i) irreducible continuous (hence $C^{\infty}$) linear representations $\rho:G(\mathbf{R}) \rightarrow {\rm{GL}}(V)$ for finite-dimensional $\mathbf{C}$-vector spaces $V$ to (ii) irreducible algebraic representations of $G'_{\mathbf{C}}$ up to twisting by continuous characters $G(\mathbf{R})/G'(\mathbf{R}) \rightarrow \mathbf{C}^{\times}$ (the familiar operation of twisting against characters applied to the determinant when $G = {\rm{GL}}_n$).
In complete generality it works out just as well as you are seeking for ${\rm{GL}}_n$ (assuming I haven't made a terrible blunder below).  This must be documented in some book on the analytic theory of Lie groups, but it seems easier to work it out than to look it up (since many books on Lie groups don't make enough conceptual contact with the algebraic theory, especially for moving between the algebraic and analytic theories over $\mathbf{R}$ and $\mathbf{C}$ all at the same time with a general $G$ and grappling with the headaches caused by disconnectedness under passage to $\mathbf{R}$-points). To formulate what is to be shown requires some preparations, as follows.
Let's fix a maximal split $\mathbf{R}$-torus $S$ in $G$ and a maximal $\mathbf{R}$-torus $T \subset G$ containing $S$, so $T' := T \cap G'$ is a maximal $\mathbf{R}$-torus of $G'$. Choose a Borel subgroup $B' \subset G'_{\mathbf{C}}$ containing $T'_{\mathbf{C}}$, and let $B \subset G_{\mathbf{C}}$ be the unique Borel subgroup meeting $G'_{\mathbf{C}}$ in $B'$ (so $T_{\mathbf{C}} \subset B$);  this defines a notion of "dominant weight" (of $T'_{\mathbf{C}}$) for irreducible algebraic representations of $G'_{\mathbf{C}}$.
We will see below that: (i) any finite-dimensional continuous $\mathbf{C}$-linear representation of $G'(\mathbf{R})$ arises uniquely via restriction from an algebraic representation of $G'_{\mathbf{C}}$ on the same vector space, and (ii) if we begin with an irreducible continuous linear representation of $G(\mathbf{R})$ on a finite-dimensional $\mathbf{C}$-vector space $V$ then it is also irreducible as a $G'(\mathbf{R})$-representation (which thus in turn uniquely extends to an irreducible algebraic representation of $G'_{\mathbf{C}}$).
The geometric construction of the highest-weight representation of $G'_{\mathbf{C}}$ corresponding to a dominant character $\chi: T'_{\mathbf{C}} \rightarrow {\rm{GL}}_1$ in terms of a line bundle on $G'_{\mathbf{C}}/B' = G_{\mathbf{C}}/B$ works in terms of $(G_{\mathbf{C}}, T_{\mathbf{C}},B)$ upon extending $\chi$ to a character of $T_{\mathbf{C}}$ (and such an extension of $\chi$ amounts to extending the central character for $G'_{\mathbf{C}}$ to a character of the center of $G_{\mathbf{C}}$, as we may always do). This defines an $G_{\mathbf{C}}$-action that restricts to a $G(\mathbf{R})$-representation extending any given irreducible continuous linear representation of $G'(\mathbf{R})$ on a finite-dimensional $\mathbf{C}$-vector space.   Also, twisting against continuous $\mathbf{C}^{\times}$-valued characters of the commutative Lie group $G(\mathbf{R})/G'(\mathbf{R})$ moves around among continuous finite-dimensional $G(\mathbf{R})$-representations with a given central character and a given restriction to $G'(\mathbf{R})$.  
We will show in a sense made precise below (needing some care with central characters on $\mathbf{R}$-points) that these two procedures exhaust all possibilities, so for $G={\rm{GL}}_n$ it yields exactly what you want.
The main battle in this generality is with disconnectedness issues for the group $G(\mathbf{R})$, and to show that irreducibility is automatically inherited by the action of the subgroup $G'(\mathbf{R})$ (not quite obvious since if $Z_G$ denotes the $\mathbf{R}$-subgroup center of $G$, so $Z_G(\mathbf{R})$ coincides with the center of $G(\mathbf{R})$ because $G(\mathbf{R})$ is Zariski-dense in the Zariski-connected $G$, then $Z_G(\mathbf{R})G'(\mathbf{R})$ might not exhaust $G(\mathbf{R})$ for disconnectedness reasons; this happens for $G={\rm{GL}}_{2m}$, for example).

To begin, consider a continuous $\mathbf{C}$-linear representation $\rho:G(\mathbf{R}) \rightarrow {\rm{GL}}(V)$ without an irreducibility assumption. Since $G'$ is simply connected, by a (hard) theorem of E. Cartan we know that $G'(\mathbf{R})$ is connected. (This is much more elementary when $G'$ is $\mathbf{R}$-split, such as ${\rm{SL}}_n$).) Thus, $\rho|_{G'(\mathbf{R})}$ is determined by its effect on Lie algebras, which is to say the induced map $\mathfrak{g}' = [\mathfrak{g}, \mathfrak{g}] \rightarrow \mathfrak{gl}(V)$ of real Lie algebras. This latter map is "the same" as a map $\mathfrak{g}'_{\mathbf{C}} \rightarrow \mathfrak{gl}(V)$ of complex Lie algebras. 
For any linear algebraic groups $H_1$ and $H_2$ over a field $k$ of characteristic 0 such that $H_1$ is connected semisimple and moreover simply connected (in the sense of connected semisimple groups), every map of Lie algebras $\mathfrak{h}_1 \rightarrow \mathfrak{h}_2$ over $k$ uniquely arises from a $k$-homomorphism $H_1 \rightarrow H_2$ (as one sees via a graph argument). Applying the latter with $k = \mathbf{C}$, $H_1 = G'_{\mathbf{C}}$, and $H_2 = {\rm{GL}}(V)$ (as a linear algebraic group over $\mathbf{C}$), we see that any continuous linear representation $G'(\mathbf{R}) \rightarrow {\rm{GL}}(V)$ arises as the restriction to $G'(\mathbf{R}) \subset G'(\mathbf{C})$ of a unique $\mathbf{C}$-homomorphism $G'_{\mathbf{C}} \rightarrow {\rm{GL}}(V)$. 
It is a general fact (due to Borel and Tits) that if $S \subset G$ is a maximal split $\mathbf{R}$-torus then $S(\mathbf{R})$ meets every connected component of $G(\mathbf{R})$. Thus, if $Z$ denotes the maximal central $\mathbf{R}$-torus in $G$ then $G(\mathbf{R})$ is generated by $S(\mathbf{R})$ and the normal subgroup $G(\mathbf{R})^0 = Z(\mathbf{R})^0 G'(\mathbf{R})$ (equality by Lie algebra reasons). In particular, if $T \subset G$ is a maximal $\mathbf{R}$-torus containing containing $S$ then $G(\mathbf{R}) = T(\mathbf{R})G'(\mathbf{R})$ since $Z, S \subset T$. Note that typically $Z(\mathbf{R})G'(\mathbf{R})$ does not exhaust $G(\mathbf{R})$ due to disconnectedness problems on $\mathbf{R}$-points (e.g., this happens for $G = {\rm{GL}}_{2m}$, for which we even have $Z=Z_G$).
Let's now assume the continuous (hence $C^{\infty}$) linear representation $\rho:G(\mathbf{R}) \rightarrow {\rm{GL}}(V)$ is  irreducible. Using the choice of Borel $\mathbf{C}$-subgroup $B'$ of $G'_{\mathbf{C}}$ containing the maximal torus $T'_{\mathbf{C}}$ for $T' := T \cap G'$, let $\chi$ be some dominant highest weight for the action of $T'_{\mathbf{C}}$ on $V$.  (We don't yet know if the action of $G'_{\mathbf{C}}$-action on $V$ is isotypic, let alone irreducible.) The $G'(\mathbf{C})$-orbits of the $\chi$-weight lines span the isotypic subspace in $V$ for the irreducible representation $\sigma$ of $G'_{\mathbf{C}}$ with highest weight $\chi$, and this span is visibly stable under the action of $G'(\mathbf{R})T(\mathbf{R})=G(\mathbf{R})$ since $T(\mathbf{R})$-conjugation on $G'(\mathbf{C})$ is trivial on $T'(\mathbf{C})$, so that span exhausts $V$ by irreducibility. In other words, $V$ is $\sigma$-isotypic for $G'_{\mathbf{C}}$.
Since the action on $V$ by the commutative $T(\mathbf{R})$ preserves the $\mathbf{C}$-span of the $\chi$-weight lines for $T'(\mathbf{C})$, its action on that span of weight lines admits a $T(\mathbf{R})$-stable line $L$.  But any such line has $G'(\mathbf{C})$-orbit that spans a copy $W \subset V$ of $\sigma$ due to the $G'(\mathbf{C})$-equivariant isomorphism $V \simeq M \otimes \sigma$ for the "multiplicity space" $M = {\rm{Hom}}_{G'(\mathbf{C})}(\sigma,V)$ (as $M$ is identified with the span of the $\chi$-weight lines, by the Theorem of the Highest Weight). 
Since $G'(\mathbf{C})T(\mathbf{R}) \supset G'(\mathbf{R})T(\mathbf{R})=G(\mathbf{R})$ inside $G(\mathbf{C})$, it follows that the subspace $W \subset V$ is $G(\mathbf{R})$-stable, forcing $W=V$ by irreducibility of $\rho$. In other words, $V$ is necessarily irreducible as a $G'(\mathbf{C})$-representation, or equivalently as a $G'(\mathbf{R})$-representation (due to Lie algebra considerations, thanks to the connectedness of $G'(\mathbf{R})$). 
The remaining issue is to determine the ways we can extend the irreducible $G'(\mathbf{R})$-action arising on an irreducible algebraic representation $(V, \sigma)$ of $G'_{\mathbf{C}}$ to a continuous representation of $G(\mathbf{R})$. 
Any possible continuous extension of $\sigma|_{G'(\mathbf{R})}$ to a $G(\mathbf{R})$-representation on $V$ must make the center $Z_G(\mathbf{R})$ of $G(\mathbf{R})$ act
through a continuous character $\psi: Z_G(\mathbf{R})\rightarrow \mathbf{C}^{\times}$ that extends the central character $Z_{G'}(\mathbf{R}) \rightarrow \mathbf{C}^{\times}$ of $\sigma|_{G'(\mathbf{R})}$. (Note that $Z_{G'} = Z_G \cap G'$, so $Z_{G'}(\mathbf{R}) = Z_G(\mathbf{R}) \cap G'(\mathbf{R})$ and this is the finite center of $G'(\mathbf{R})$.)  Such a $\psi$ always exists, one such being given by the "holomorphic" (or "$\mathbf{C}$-algebraic") construction using flag varieties mentioned at the outset.
Any two such $\psi$'s are related through multiplication against a continuous $\mathbf{C}^{\times}$-valued character of the finite-index open subgroup $Z_G(\mathbf{R})/Z_{G'}(\mathbf{R})$ of $G(\mathbf{R})/G'(\mathbf{R})$, so they are related through multiplication against the restriction of a continuous $\mathbf{C}^{\times}$-valued character of $G(\mathbf{R})/G'(\mathbf{R})$. 
Hence, since we are aiming to describe all possibilities modulo the effect of twisting against a continuous $\mathbf{C}^{\times}$-valued character of $G(\mathbf{R})/G'(\mathbf{R})$, it suffices to fix a choice of $\psi$ and show that up to isomorphism there is only one continuous extension of $\sigma|_{G'(\mathbf{R})}$ to a linear $G(\mathbf{R})$-action on $V$ with central character $\psi$.  
By Lie algebra considerations clearly $G(\mathbf{R})^0 \subset Z_G(\mathbf{R}) G'(\mathbf{R})$, and $S(\mathbf{R}) = S(\mathbf{R})[2] \times S(\mathbf{R})^0$ since $S$ is a split $\mathbf{R}$-torus, we have 
$$G(\mathbf{R}) = S(\mathbf{R})[2] \cdot (Z_G(\mathbf{R})G'(\mathbf{R}))$$
with $S(\mathbf{R})[2]$ a finite abelian subgroup of $T(\mathbf{R})$.
By design of $\psi$, the action of $G'(\mathbf{R})$ on $V$ uniquely extends to an action on $V$ by the finite-index open normal subgroup
$$N := Z_G(\mathbf{R}) G'(\mathbf{R}) = Z_G(\mathbf{R}) \times^{Z_{G'}(\mathbf{R})} G'(\mathbf{R})$$
of $G(\mathbf{R})$ with central character $\psi$, and this extension is visibly continuous; let's write $V_{\psi}$ to denote $V$ with this extended group action. Our task is now purely algebraic: show that up to isomorphism this admits only one extension to a linear action of $G(\mathbf{R})$ on $V_{\psi}$.
Since $G(\mathbf{R})/N$ is represented (with possible repetitions) by the (finite) subgroup $S(\mathbf{R})[2]$ of $G(\mathbf{R})$ that trivially commutes with $Z_G(\mathbf{R})$ and has conjugation action on $G'_{\mathbf{C}}$ that is trivial on $T'_{\mathbf{C}}$, when we view the finite-dimensional (continuous!) representation
$${\rm{Ind}}_{N}^{G(\mathbf{R})}(V_{\psi})$$
of $G(\mathbf{R})$ as an $N$-representation, it is (non-canonically) a direct sum of $$n:=[G(\mathbf{R}):N]$$ copies of $V_{\psi}$ due to the Theorem of the Highest Weight.  This induction is also semisimple as a $G(\mathbf{R})$-representation because $V_{\psi}$ is a semisimple $N$-representation with $N$ normal of finite index in $G(\mathbf{R})$. 
Writing the induction as a direct sum of finitely many irreducible $G(\mathbf{R})$-representations, the number of such summands is $n$ because its irreducible $G(\mathbf{R})$-subrepresentations are precisely its irreducible $G'(\mathbf{R})$-subrepresentations due to the fact (which we have seen earlier) that every irreducible continuous finite-dimensional $G(\mathbf{R})$-representation remains irreducible for $G'(\mathbf{R})$. Thus, by semisimplicity, the $G(\mathbf{R})$-stable subspaces  of ${\rm{Ind}}_N^{G(\mathbf{R})}(V_{\psi})$ are precisely the $N$-stable subspaces of its underlying $N$-representation that in turn is isomorphic to $V_{\psi}^{\oplus n}$, and likewise with $N$ replaced by $G'(\mathbf{R})$ (noting that the $N$-representation $V_{\psi}$ has underlying $G'(\mathbf{R})$-representation $V$). Hence, by Schur's Lemma we have as $\mathbf{C}$-algebras
$${\rm{End}}_{G(\mathbf{R})}({\rm{Ind}}_N^{G(\mathbf{R})}(V_{\psi}))={\rm{End}}_{G'(\mathbf{R})}(V^{\oplus n}) = {\rm{Mat}}_n(\mathbf{C}).$$
It follows that the semisimple $G(\mathbf{R})$-representation  ${\rm{Ind}}_N^{G(\mathbf{R})}(V_{\psi})$ is isomorphic to a direct sum of $n$ copies of an irreducible $G(\mathbf{R})$-action on $V_{\psi}$ extending the $N$-action. This establishes the desired uniqueness up to isomorphism as a $G(\mathbf{R})$-representation, so we are done.
