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Perhaps it should be noted that this is a more genral phenomena:

The Schur lemma tells you that the restriction of an irreducible representation of a locally compact group $G$ to the centrum $Z$ is always isomorphic to a character. Hence by a twist of character, we can always assume without loss of generality that the central character is trivial and study the representation theory of $G/Z$ only.

In the case you mention, this observation is important since $GL_2(F) \backslash GL_2(A)$ has not finite volume in the canonical measure, but $GL_2(F) Z(A) \backslash GL_2(A)$ does. This is a necessary point, if you want to analyse the representations via the Arthur trace formula, which does make sense for finite volume quotients only.

Maass (cusp) forms and modular (cusp) forms give both (cuspidal) automorphic representations and the cuspidal ones give vectors of $L_0$. The main point is the observation: $$ GL_2(\mathbb{Q}) z(A) \backslash GL_2(A) /\prod_p GL_2(\mathbb{Z}_p) O(2)$$ $$= PSL_2( \mathbb{Z}) \backslash \mathbb{H}.$$ An explanation, how to see this, is in Gelbarts "Adeles Groups ... " book, if I recall it correctly,and there is an article of Kudla in Cogdell et al. "Introduction to Langlands program". I think you should consider these only as different objects, if you want argue with algebraic geometry, which applies to holomorphic stuff only, but not if you want to do representation theory. Edit due to the comments: Maass forms and modular forms seen as representations look different at the archimedean prime, i.e. have a "different" representation of $GL_2(\mathbb{R})$ there.

Perhaps it should be noted that this is a more genral phenomena:

The Schur lemma tells you that the restriction of an irreducible representation of a locally compact group $G$ to the centrum $Z$ is always isomorphic to a character.

Assummin fixed central character, is a technical requirement, but important since $GL_2(F) \backslash GL_2(A)$ has not finite quotient measure, but $GL_2(F) Z(A) \backslash GL_2(A)$ does. Alternatively, you can look $GL_2(A)^1$, i.e. the kernel $g \mapsto | \det g|_A$, since $GL_2(F) \backslash GL_2(A)^1$ has finite volume as well.

This is a necessary point, if you want to analyse the representations via the Arthur trace formula, which does make sense for finite volume quotients only.

Maass (cusp) forms and modular (cusp) forms give both (cuspidal) automorphic representations and the cuspidal ones give vectors of $L_0$. The main point is the observation: $$ GL_2(\mathbb{Q}) z(A) \backslash GL_2(A) /\prod_p GL_2(\mathbb{Z}_p) O(2)$$ $$= PSL_2( \mathbb{Z}) \backslash \mathbb{H}.$$ An explanation, how to see this, is in Gelbarts "Adeles Groups ... " book, if I recall it correctly,and there is an article of Kudla in Cogdell et al. "Introduction to Langlands program". I think you should consider these only as different objects, if you want argue with algebraic geometry, which applies to holomorphic stuff only, but not if you want to do representation theory. Edit due to the comments: Maass forms and modular forms seen as representations look different at the archimedean primes, i.e. have a "different" representation of $GL_2(\mathbb{R})$ there, i.e. prinicpal series vs. discrete series.

Perhaps it should be noted that this is a more genral phenomena:

The Schur lemma tells you that the restriction of an irreducible representation of a locally compact group $G$ to the centrum $Z$ is always isomorphic to a character. Hence by a twist of character, we can always assume without loss of generality that the central character is trivial and study the representation theory of $G/Z$ only.

In the case you mention, this observation is important since $GL_2(F) \backslash GL_2(A)$ has not finite volume in the canonical measure, but $GL_2(F) Z(A) \backslash GL_2(A)$ does. This is a necessary point, if you want to analyse the representations via the Arthur trace formula, which does make sense for finite volume quotients only.

Maass (cusp) forms and modular (cusp) forms give both (cuspidal) automorphic representations and the cuspidal ones give vectors of $L_0$. The main point is the observation: $$ GL_2(\mathbb{Q}) Z(A) \backslash GL_2(A) /\prod_p GL_2(\mathbb{Z}_p) O(2)$$ $$= SL_2( \mathbb{Z}) \backslash \mathbb{H}.$$ An explanation, how to see this, is in Gelbarts "Adeles Groups ... " book, if I recall it correctly,and there is an article of Kudla in Cogdell et al. "Introduction to Langlands program". I think you should consider these only as different objects, if you want argue with algebraic geometry, which applies to holomorphic stuff only, but not if you want to do representation theory.

Perhaps it should be noted that this is a more genral phenomena:

The Schur lemma tells you that the restriction of an irreducible representation of a locally compact group $G$ to the centrum $Z$ is always isomorphic to a character. Hence by a twist of character, we can always assume without loss of generality that the central character is trivial and study the representation theory of $G/Z$ only.

In the case you mention, this observation is important since $GL_2(F) \backslash GL_2(A)$ has not finite volume in the canonical measure, but $GL_2(F) Z(A) \backslash GL_2(A)$ does. This is a necessary point, if you want to analyse the representations via the Arthur trace formula, which does make sense for finite volume quotients only.

Maass (cusp) forms and modular (cusp) forms give both (cuspidal) automorphic representations and the cuspidal ones give vectors of $L_0$. The main point is the observation: $$ GL_2(\mathbb{Q}) z(A) \backslash GL_2(A) /\prod_p GL_2(\mathbb{Z}_p) O(2)$$ $$= PSL_2( \mathbb{Z}) \backslash \mathbb{H}.$$ An explanation, how to see this, is in Gelbarts "Adeles Groups ... " book, if I recall it correctly,and there is an article of Kudla in Cogdell et al. "Introduction to Langlands program". I think you should consider these only as different objects, if you want argue with algebraic geometry, which applies to holomorphic stuff only, but not if you want to do representation theory. Edit due to the comments: Maass forms and modular forms seen as representations look different at the archimedean prime, i.e. have a "different" representation of $GL_2(\mathbb{R})$ there.

Perhaps it should be noted that this is a more genral phenomena:

The Schur lemma tells you that the restriction of an irreducible representation of a locally compact group $G$ to the centrum $Z$ is always isomorphic to a character. Hence by a twist of character, we can always assume without loss of generality that the central character is trivial and study the representation theory of $G/Z$ only.

In the case you mention, this observation is important since $GL_2(F) \backslash GL_2(A)$ has not finite volume in the canonical measure, but $GL_2(F) Z(A) \backslash GL_2(A)$ does. This is a necessary point, if you want to analyse the representations via the Arthur trace formula, which does make sense for finite volume quotients only.

Maass (cusp) forms and modular (cusp) forms give both (cuspidal) automorphic representations and the cuspidal ones give vectors of $L_0$. An explanation, how to see this, is in Gelbarts "Adeles Groups ... " book, if I recall it correctly,and there is an article of Kudla in Cogdell et al. "Introduction to Langlands program". I think you should consider these only as different objects, if you want argue with algebraic geometry, which applies to holomorphic stuff only, but not if you want to do representation theory.

Perhaps it should be noted that this is a more genral phenomena:

The Schur lemma tells you that the restriction of an irreducible representation of a locally compact group $G$ to the centrum $Z$ is always isomorphic to a character. Hence by a twist of character, we can always assume without loss of generality that the central character is trivial and study the representation theory of $G/Z$ only.

In the case you mention, this observation is important since $GL_2(F) \backslash GL_2(A)$ has not finite volume in the canonical measure, but $GL_2(F) Z(A) \backslash GL_2(A)$ does. This is a necessary point, if you want to analyse the representations via the Arthur trace formula, which does make sense for finite volume quotients only.

Maass (cusp) forms and modular (cusp) forms give both (cuspidal) automorphic representations and the cuspidal ones give vectors of $L_0$. The main point is the observation: $$ GL_2(\mathbb{Q}) Z(A) \backslash GL_2(A) /\prod_p GL_2(\mathbb{Z}_p) O(2)$$ $$= SL_2( \mathbb{Z}) \backslash \mathbb{H}.$$ An explanation, how to see this, is in Gelbarts "Adeles Groups ... " book, if I recall it correctly,and there is an article of Kudla in Cogdell et al. "Introduction to Langlands program". I think you should consider these only as different objects, if you want argue with algebraic geometry, which applies to holomorphic stuff only, but not if you want to do representation theory.