1 wavenumber shift isn't much - it's on the order of the instrument resolution and far narrower than most carbonyl stretch bandwidths - but may be enough.
The Diels Alder adduct of maleic anhydride and anthracene forms something that closely resembles succinic anhydride (like maleic anhydride without the double bond). Anhydrides in general have two carbonyl resonances in the 1700-1875 wavenumber region. These stretches are coupled - one is a symmetric stretch and one is an asymmetric stretch. Carbonyl stretches of 5-member cyclic anhydrides usually vibrate at higher frequencies due to strain in the ring. Succinic anhydride (plain) has absorption at 1865 and 1782 wavenumbers. Compare this to acetic anhydride, which has streches at around 1818 and 1750 wavenumbers. The symmetric stretch is the one at lower frequency.
I tried but couldn't find actual stretching information for maleic anhydride and the spectrum posted at NIST or aldrich aren't particularly helpful.
http://webbook.nist.gov/cgi/cbook.cgi?ID=C108316&Type=IR-SPEC&Index=0#IR-SPEChttp://www.sigmaaldrich.com/spectra/ftir/FTIR008271.PDFFrom the latter it appears maleic anhydride also has two carbonyl stretches in roughly the same position.
Conjugation to a carbonyl reduces the double bond character and causes the carbonyl to shift to lower frequencies. In acylic anhydrides this effect is pretty substantial (30 to 40 wavenumbers), but it seems the addition of a double bond does not impact much the position of the stretches in 5-membered cyclic anhydrides. It is possible this is because the ring strain is by far the dominant effect. Even so the 1 wavenumber shift to higher frequency MAY be indicative of a loss of the double bond in maleic anhydride when formation of the adduct occurs. However this is so small I wouldn't use this alone.
The other area you may want to look is the double bond itself. C=C stretches are usually fairly weak, but the double bond here is cis-orientation which will help. Typically the C=C stretch shows up in the vicinity of 1600 to 1650 wavenumbers. In a strained system, the shift is again shifted to lower frequencies (to a point). From the NIST/Adrich spectra it appears that the reasonably sharp, intense peak at approximately 1595 wavenumbers may be due to this C=C stretch - but you should be able to provide more specific value from your data. In the adduct, this peak should disappear because there's no longer a double bond to stretch. Do you see it go away?
The other area you may see a difference is in the aromatic stretching modes, because anthracene is no longer anthracene once the adduct forms, but this may be trick.
Really what you need to do is take an IR spectrum of an anthracene/maleic anhydride mixture (or separately, and just superimpose them) compared to the product. There should be sufficient differences to see whether the product spectrum is just an overlay of the reactant spectra, or whether there are new peaks not seen in either the reactant spectra. Once you identify new peaks, then you can work to identify what they are from.