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paul garrett
paul garrett
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(I took @LSpice's inquiry as encouragement to add some remarks to @GH from MO's good answer... But/and one of the issues that helped me overcome my skepticism about the utility of representation theory long ago was its clarification of exactly these issues about "Hecke summation", Maass-Shimura operators, and such. In particular, that it is not necessary to write out Fourier expansions using special functions, etc.)

First, as in the comments that helped @GH...'s answer get on track, in a typical (naive) indexing scheme, the $2k$-th holomorphic discrete series is not a subrepresentation of the $2k$-th principal series (also in a naive normalization), but just of the $k$-th. Also, there are choices about when-and-how to append/renormalize the $y^k$ to weight-$2k$ automorphic forms, to make them left $\Gamma$-invariant and right $K$-equivariant. Lots of chances to make a mess...

A more interesting issue than normalization is the fact that (the map...) formation of Eisenstein series (manifestly) does not commute with meromorphic continuation. Such a thing already occurs in looking at the spectral decomposition of the space of pseudo-Eisenstein series in terms of Eisenstein series.

For generic principal series $I_s$, the raising and lowering operators $R,L$ change the $K$-type by $\pm 2$, with the lowering operator having a coefficient of $s-k$ (here the normalization matters, obviously). So, up to irrelevant constants, with $E_{s,2k}$ being the Eisenstein series hit by $I_s$ with $K$-type $2k$, $LE_{s,2k}=(s-1)E_{s,2k-2}$$LE_{s,2k}=(s-k)E_{s,2k-2}$. Thus, at $s=1$, if $E_{s,2k-2}$ has no pole, then $E_{s,2k}$ is annihilated by the lowering operator.

It is partly some sort of crazy luck that the lowering operator $L$, in coordinates, is essentially the Cauchy-Riemann operator, so annihilation by it implies holomorphy.

But/and of course with $2k=2$, the Eisenstein series $E_{s,0}$ has a pole at $s=1$. The residue is just a constant, not very exciting, but not $0$. Thus, in whatever normalization one operates, the application of the lowering operator (Cauchy-Riemann) to (the meromorphically continued) $E_{s,2}$ gives that residue, not $0$.

For Hilbert modular forms over (totally real) number fields of degree $>1$ over $\mathbb Q$, the lowering operators attached to the various archimedean factors of the group do not map $E_{s,(2,2,2,2,...)}$ to $E_{s,0}$, where there is a pole, so they annihilate $E_{s,(2,2,...)}$.

The weight-one Eisenstein series are another story...

(Edit: some "$s-1$"'s should have been "$s-k$"'s, but I was thinking about $k=1$... maybe repaired...)

(I took @LSpice's inquiry as encouragement to add some remarks to @GH from MO's good answer... But/and one of the issues that helped me overcome my skepticism about the utility of representation theory long ago was its clarification of exactly these issues about "Hecke summation", Maass-Shimura operators, and such. In particular, that it is not necessary to write out Fourier expansions using special functions, etc.)

First, as in the comments that helped @GH...'s answer get on track, in a typical (naive) indexing scheme, the $2k$-th holomorphic discrete series is not a subrepresentation of the $2k$-th principal series (also in a naive normalization), but just of the $k$-th. Also, there are choices about when-and-how to append/renormalize the $y^k$ to weight-$2k$ automorphic forms, to make them left $\Gamma$-invariant and right $K$-equivariant. Lots of chances to make a mess...

A more interesting issue than normalization is the fact that (the map...) formation of Eisenstein series (manifestly) does not commute with meromorphic continuation. Such a thing already occurs in looking at the spectral decomposition of the space of pseudo-Eisenstein series in terms of Eisenstein series.

For generic principal series $I_s$, the raising and lowering operators $R,L$ change the $K$-type by $\pm 2$, with the lowering operator having a coefficient of $s-k$ (here the normalization matters, obviously). So, up to irrelevant constants, with $E_{s,2k}$ being the Eisenstein series hit by $I_s$ with $K$-type $2k$, $LE_{s,2k}=(s-1)E_{s,2k-2}$. Thus, at $s=1$, if $E_{s,2k-2}$ has no pole, then $E_{s,2k}$ is annihilated by the lowering operator.

It is partly some sort of crazy luck that the lowering operator $L$, in coordinates, is essentially the Cauchy-Riemann operator, so annihilation by it implies holomorphy.

But/and of course with $2k=2$, the Eisenstein series $E_{s,0}$ has a pole at $s=1$. The residue is just a constant, not very exciting, but not $0$. Thus, in whatever normalization one operates, the application of the lowering operator (Cauchy-Riemann) to (the meromorphically continued) $E_{s,2}$ gives that residue, not $0$.

For Hilbert modular forms over (totally real) number fields of degree $>1$ over $\mathbb Q$, the lowering operators attached to the various archimedean factors of the group do not map $E_{s,(2,2,2,2,...)}$ to $E_{s,0}$, where there is a pole, so they annihilate $E_{s,(2,2,...)}$.

The weight-one Eisenstein series are another story...

(I took @LSpice's inquiry as encouragement to add some remarks to @GH from MO's good answer... But/and one of the issues that helped me overcome my skepticism about the utility of representation theory long ago was its clarification of exactly these issues about "Hecke summation", Maass-Shimura operators, and such. In particular, that it is not necessary to write out Fourier expansions using special functions, etc.)

First, as in the comments that helped @GH...'s answer get on track, in a typical (naive) indexing scheme, the $2k$-th holomorphic discrete series is not a subrepresentation of the $2k$-th principal series (also in a naive normalization), but just of the $k$-th. Also, there are choices about when-and-how to append/renormalize the $y^k$ to weight-$2k$ automorphic forms, to make them left $\Gamma$-invariant and right $K$-equivariant. Lots of chances to make a mess...

A more interesting issue than normalization is the fact that (the map...) formation of Eisenstein series (manifestly) does not commute with meromorphic continuation. Such a thing already occurs in looking at the spectral decomposition of the space of pseudo-Eisenstein series in terms of Eisenstein series.

For generic principal series $I_s$, the raising and lowering operators $R,L$ change the $K$-type by $\pm 2$, with the lowering operator having a coefficient of $s-k$ (here the normalization matters, obviously). So, up to irrelevant constants, with $E_{s,2k}$ being the Eisenstein series hit by $I_s$ with $K$-type $2k$, $LE_{s,2k}=(s-k)E_{s,2k-2}$. Thus, at $s=1$, if $E_{s,2k-2}$ has no pole, then $E_{s,2k}$ is annihilated by the lowering operator.

It is partly some sort of crazy luck that the lowering operator $L$, in coordinates, is essentially the Cauchy-Riemann operator, so annihilation by it implies holomorphy.

But/and of course with $2k=2$, the Eisenstein series $E_{s,0}$ has a pole at $s=1$. The residue is just a constant, not very exciting, but not $0$. Thus, in whatever normalization one operates, the application of the lowering operator (Cauchy-Riemann) to (the meromorphically continued) $E_{s,2}$ gives that residue, not $0$.

For Hilbert modular forms over (totally real) number fields of degree $>1$ over $\mathbb Q$, the lowering operators attached to the various archimedean factors of the group do not map $E_{s,(2,2,2,2,...)}$ to $E_{s,0}$, where there is a pole, so they annihilate $E_{s,(2,2,...)}$.

The weight-one Eisenstein series are another story...

(Edit: some "$s-1$"'s should have been "$s-k$"'s, but I was thinking about $k=1$... maybe repaired...)

Source Link
paul garrett
paul garrett
  • 23k
  • 3
  • 86
  • 125

(I took @LSpice's inquiry as encouragement to add some remarks to @GH from MO's good answer... But/and one of the issues that helped me overcome my skepticism about the utility of representation theory long ago was its clarification of exactly these issues about "Hecke summation", Maass-Shimura operators, and such. In particular, that it is not necessary to write out Fourier expansions using special functions, etc.)

First, as in the comments that helped @GH...'s answer get on track, in a typical (naive) indexing scheme, the $2k$-th holomorphic discrete series is not a subrepresentation of the $2k$-th principal series (also in a naive normalization), but just of the $k$-th. Also, there are choices about when-and-how to append/renormalize the $y^k$ to weight-$2k$ automorphic forms, to make them left $\Gamma$-invariant and right $K$-equivariant. Lots of chances to make a mess...

A more interesting issue than normalization is the fact that (the map...) formation of Eisenstein series (manifestly) does not commute with meromorphic continuation. Such a thing already occurs in looking at the spectral decomposition of the space of pseudo-Eisenstein series in terms of Eisenstein series.

For generic principal series $I_s$, the raising and lowering operators $R,L$ change the $K$-type by $\pm 2$, with the lowering operator having a coefficient of $s-k$ (here the normalization matters, obviously). So, up to irrelevant constants, with $E_{s,2k}$ being the Eisenstein series hit by $I_s$ with $K$-type $2k$, $LE_{s,2k}=(s-1)E_{s,2k-2}$. Thus, at $s=1$, if $E_{s,2k-2}$ has no pole, then $E_{s,2k}$ is annihilated by the lowering operator.

It is partly some sort of crazy luck that the lowering operator $L$, in coordinates, is essentially the Cauchy-Riemann operator, so annihilation by it implies holomorphy.

But/and of course with $2k=2$, the Eisenstein series $E_{s,0}$ has a pole at $s=1$. The residue is just a constant, not very exciting, but not $0$. Thus, in whatever normalization one operates, the application of the lowering operator (Cauchy-Riemann) to (the meromorphically continued) $E_{s,2}$ gives that residue, not $0$.

For Hilbert modular forms over (totally real) number fields of degree $>1$ over $\mathbb Q$, the lowering operators attached to the various archimedean factors of the group do not map $E_{s,(2,2,2,2,...)}$ to $E_{s,0}$, where there is a pole, so they annihilate $E_{s,(2,2,...)}$.

The weight-one Eisenstein series are another story...