The quantum notation used in the "traditional" .par formatted line-by-line lists for each molecule in the HITRAN database was described in detail in the supplementary material of the HITRAN2020 paper [1]. This information is reproduced in the following document, and will be maintained for future updates to the database:

In addition, supplementary XSAMS notation (with more detail) is available for each line if one requests a custom output format to include this information. Details of the XSAMS notation is available on the Virtual Atomic and Molecular Database Consortium (VAMDC) website.
 

References

[1] I. E. Gordon, L. S. Rothman, R. J. Hargreaves, R. Hashemi, E. V. Karlovets, F. M. Skinner, et al., "The HITRAN2020 molecular spectroscopic database", J. Quant. Spectrosc. Radiat. Transfer 277, 107949 (2022). [link to article]