Relationship between scalar and vector valued automorphic forms

Question

Let $F$ be a number field with signature $(r_1,r_2)$. There are two definitions of automorphic forms on $GL_{2,F}$ that I am aware of. Let $K=O_2(\mathbb R)^{r_1}\times U_2(\mathbb C)^{r_2}$ be a maximal compact subgroup of $GL_2(F)$. Roughly speaking, the two definitions are

Definition 1 (scalar valued): An automorphic form is a smooth function $f:GL_2(\mathbb A_F)\to \mathbb C$ that is left $GL_2(F)$-invariant, right $K$ finite, ... Denote the space of all such functions by $\mathcal A$.

Definition 2 (vector valued): Fix an irreducible representation $(\rho,V)$ of $K$. An automorphic form (of weight $V$) is a smooth function $f:GL_2(\mathbb A_F)\to V$ that is left $GL_2(F)$-invariant, $f(gk)=\rho(k)^{-1}f(g)$ for all $g\in GL_2(\mathbb A_F),k\in K,$... Denote the space of all such functions by $\mathcal A(\rho)$.

May I ask what is the relationship between these two notions? For instance, will a statement like $\mathcal A\cong \oplus_{\rho} \mathcal A(\rho)\otimes_{\mathbb C} V^\vee$, where $\rho$ runs over the isomorphism classes of finite dimensional irreducible representations of $K$, holds?

Also, given $f\in \mathcal A(\rho)$, how should one define the automorphic representation (which should be an irreducible subquotient of some space of scalar valued functions) associated to it?

Answer 1

Just to add some of my finding on this question. I am a beginner in automorphic form, so if there is any misunderstanding feel free to point it out.

Firstly, scalar valued form should be a special case of vector valued form when the associated representation of the compact subgroup $(\rho, K)$ is one-dimensional.

Secondly, from a vector-valued form we can construct a scalar-valued form by taking scalar product with a fixed vector $w\in W$. On the other hand, from a finite dimensional subspace $V$ of scalar valued forms that is stable under $K$ (not the entire group $G$ which gives rise to infinite-dimensional automorphic representation!), we get a $V$-valued vector form. This remark is from Corvallis I page 190, remark 1.5 (2).

Finally, for which I think it is the most interesting question is how do we explicitly construct vector-valued form (of dimension >1) from scalar-valued forms. Here is one method I learned from 2022 Arizona Winter school. Starting from a scalar-valued Siegel modular form (functions on the Siegel upper half plane $H_g$ that satisfies the same transformation law as modular form), we can consider the restriction onto the diagonally embedded pairs of lower-dimensional Siegel upper half planes $H_{j}\times H_{g-j}\hookrightarrow H_g$. If the restrictions is identically zero, then we can look at the multivariate taylor series expansion with variables $\{z_{(a,b)}\}_{1\le a\le j, 1\le b\le g-j}$ (the off-diagonal block), then the lowest order term in the taylor expansion gives a section of $(L_j^k\boxtimes L_{g-j}^k)\otimes \mathrm{Sym}^r(N^\vee)$ on $\Gamma_j\setminus H_j\times \Gamma_{g-j}\setminus H_{g-j}$ where $N^\vee$ is the conormal bundle on $\Gamma_j\setminus H_j\times \Gamma_{g-j}\setminus H_{g-j}$ (more precisely $N^\vee:=\mathbb E_j\boxtimes \mathbb E_{g-j}$ where $\mathbb E_{j}$ is the Hodge bundle on $\Gamma_j\setminus H_j$) and $L_j^k=\det(N^\vee)^{\otimes k}$ on $\Gamma_j\setminus H_j$ (so its sections are weight $k$ modular forms on $H_j$). This will give us a vector-valued form on the product of two symplectic groups. For precise details see this paper by Fabien Clery and Van Der Geer.