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This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms unconditionally. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.

Added. A similar question was asked at MO already, see here.

This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms unconditionally. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.

Added. A similar question was asked at MO already, see here.

This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms unconditionally. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.

This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms unconditionally. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.

Added. A similar question was asked at MO already, see here.

This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.

This is a great question. I am no expert, but I share what I know.

We conjecture that this is a bijection. The most relevant publication that I am aware of is due to Blasius-Ramakrishnan (MR1012167) who provided a strategy to associate $2$-dimensional even complex Galois representations to Maass forms with Laplace eigenvalue $1/4$. Gelbart wrote in his AMS review to this paper that subsequently the authors together with Clozel and Harris proved the algebraicity of Hecke eigenvalues of such Maass forms unconditionally. Henniart gave a Bourbaki seminar on this topic (MR1040577), and then he published an erratum (MR1157812) whose AMS review by Zink reads:

The theorem of Blasius, Clozel and Ramakrishnan that the eigenvalues of Hecke operators for Maass forms of Galois type are algebraic numbers, which the author had discussed in the original paper, has to be considered unproved up to now. The problem is that the transfer $\Pi$ of a Maass representation $\pi$ from $\mathrm{GL}_2(\mathbf{A}_\mathbf{Q})$ to $\mathrm{GSp}_4(\mathbf{A}_\mathbf{Q})$ is unlike what has been stated before, namely its infinite component $\Pi_\infty$ occurs in an $L$-packet which does not contain limits of the discrete series.

The erratum is based on a letter by Blasius and Ramakrishnan, dated 9 April 1991.